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class="vector-toc-list"> <li id="toc-Types_of_loadings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Types_of_loadings"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Types of loadings</span> </div> </a> <ul id="toc-Types_of_loadings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stress_terms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stress_terms"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Stress terms</span> </div> </a> <ul id="toc-Stress_terms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stress_parameters_for_resistance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stress_parameters_for_resistance"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Stress parameters for resistance</span> </div> </a> <ul id="toc-Stress_parameters_for_resistance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Strain_parameters_for_resistance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Strain_parameters_for_resistance"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Strain parameters for resistance</span> </div> </a> <ul id="toc-Strain_parameters_for_resistance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stress–strain_relations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stress–strain_relations"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Stress–strain relations</span> </div> </a> <ul id="toc-Stress–strain_relations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Design_terms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Design_terms"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Design terms</span> </div> </a> <button aria-controls="toc-Design_terms-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 Design terms subsection</span> </button> <ul id="toc-Design_terms-sublist" class="vector-toc-list"> <li id="toc-Failure_theories" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Failure_theories"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Failure theories</span> </div> </a> <ul id="toc-Failure_theories-sublist" class="vector-toc-list"> </ul> </li> </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">3</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">4</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div 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">Strength of materials</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. Available in 53 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-53" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">53 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%82%D8%A7%D9%88%D9%85%D8%A9_%D8%A7%D9%84%D9%85%D9%88%D8%A7%D8%AF" title="مقاومة المواد – Arabic" lang="ar" hreflang="ar" data-title="مقاومة المواد" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Resistencia_de_materiales" title="Resistencia de materiales – Asturian" lang="ast" hreflang="ast" data-title="Resistencia de materiales" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Materiallar_m%C3%BCqavim%C9%99ti" title="Materiallar müqaviməti – Azerbaijani" lang="az" hreflang="az" data-title="Materiallar müqaviməti" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%89%E0%A6%AA%E0%A6%BE%E0%A6%A6%E0%A6%BE%E0%A6%A8%E0%A7%87%E0%A6%B0_%E0%A6%B8%E0%A6%B9%E0%A6%A4%E0%A6%BE%E0%A6%AE%E0%A6%BE%E0%A6%A4%E0%A7%8D%E0%A6%B0%E0%A6%BE" title="উপাদানের সহতামাত্রা – Bangla" lang="bn" hreflang="bn" data-title="উপাদানের সহতামাত্রা" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D1%83%D0%BF%D1%80%D0%B0%D1%86%D1%96%D1%9E%D0%BB%D0%B5%D0%BD%D0%BD%D0%B5_%D0%BC%D0%B0%D1%82%D1%8D%D1%80%D1%8B%D1%8F%D0%BB%D0%B0%D1%9E" title="Супраціўленне матэрыялаў – Belarusian" lang="be" hreflang="be" data-title="Супраціўленне матэрыялаў" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D1%8A%D0%BF%D1%80%D0%BE%D1%82%D0%B8%D0%B2%D0%BB%D0%B5%D0%BD%D0%B8%D0%B5_%D0%BD%D0%B0_%D0%BC%D0%B0%D1%82%D0%B5%D1%80%D0%B8%D0%B0%D0%BB%D0%B8%D1%82%D0%B5" title="Съпротивление на материалите – Bulgarian" lang="bg" hreflang="bg" data-title="Съпротивление на материалите" 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/Resist%C3%A8ncia_de_materials" title="Resistència de materials – Catalan" lang="ca" hreflang="ca" data-title="Resistència de materials" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D1%80%D0%B8%D0%B0%D0%BB%D1%81%D0%B5%D0%BD_%D1%87%C4%83%D1%82%C4%83%D0%BC%D0%BB%C4%83%D1%85%C4%95" title="Материалсен чăтăмлăхĕ – Chuvash" lang="cv" hreflang="cv" data-title="Материалсен чăтăмлăхĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Pevnost_(fyzika)" title="Pevnost (fyzika) – Czech" lang="cs" hreflang="cs" data-title="Pevnost (fyzika)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Festigkeitslehre" title="Festigkeitslehre – German" lang="de" hreflang="de" data-title="Festigkeitslehre" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Tugevus%C3%B5petus" title="Tugevusõpetus – Estonian" lang="et" hreflang="et" data-title="Tugevusõpetus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BD%CF%84%CE%BF%CF%87%CE%AE_%CF%84%CF%89%CE%BD_%CF%85%CE%BB%CE%B9%CE%BA%CF%8E%CE%BD" title="Αντοχή των υλικών – Greek" lang="el" hreflang="el" data-title="Αντοχή των υλικών" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Resistencia_de_materiales" title="Resistencia de materiales – Spanish" lang="es" hreflang="es" data-title="Resistencia de materiales" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Mekaniko_de_materialoj" title="Mekaniko de materialoj – Esperanto" lang="eo" hreflang="eo" data-title="Mekaniko de materialoj" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Materialen_erresistentzia" title="Materialen erresistentzia – Basque" lang="eu" hreflang="eu" data-title="Materialen erresistentzia" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%82%D8%A7%D9%88%D9%85%D8%AA_%D9%85%D9%88%D8%A7%D8%AF" title="مقاومت مواد – Persian" lang="fa" hreflang="fa" data-title="مقاومت مواد" 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/R%C3%A9sistance_des_mat%C3%A9riaux" title="Résistance des matériaux – French" lang="fr" hreflang="fr" data-title="Résistance des matériaux" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Righneas_(%C3%A1bhair)" title="Righneas (ábhair) – Irish" lang="ga" hreflang="ga" data-title="Righneas (ábhair)" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-inh mw-list-item"><a href="https://inh.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D1%80%D0%B8%D0%B0%D0%BB%D0%B0%D1%88%D0%B0_%D0%B4%D1%83%D1%85%D1%8C%D0%B0%D0%BB%D0%B5_%D1%8F%D1%80" title="Материалаша духьале яр – Ingush" lang="inh" hreflang="inh" data-title="Материалаша духьале яр" data-language-autonym="ГӀалгӀай" data-language-local-name="Ingush" class="interlanguage-link-target"><span>ГӀалгӀай</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9E%AC%EB%A3%8C%EC%97%AD%ED%95%99" title="재료역학 – Korean" lang="ko" hreflang="ko" data-title="재료역학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%86%D5%B5%D5%B8%D6%82%D5%A9%D5%A5%D6%80%D5%AB_%D5%A4%D5%AB%D5%B4%D5%A1%D5%A4%D6%80%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Նյութերի դիմադրություն – Armenian" lang="hy" hreflang="hy" data-title="Նյութերի դիմադրություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AA%E0%A4%A6%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%A5_%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%AC%E0%A4%B2%E0%A4%A4%E0%A4%BE" title="पदार्थ प्रबलता – Hindi" lang="hi" hreflang="hi" data-title="पदार्थ प्रबलता" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Nauka_o_%C4%8Dvrsto%C4%87i" title="Nauka o čvrstoći – Croatian" lang="hr" hreflang="hr" data-title="Nauka o čvrstoći" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Kekuatan_bahan" title="Kekuatan bahan – Indonesian" lang="id" hreflang="id" data-title="Kekuatan bahan" 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/Resistenza_meccanica" title="Resistenza meccanica – Italian" lang="it" hreflang="it" data-title="Resistenza meccanica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%95%D7%96%D7%A7_%D7%97%D7%95%D7%9E%D7%A8%D7%99%D7%9D" title="חוזק חומרים – Hebrew" lang="he" hreflang="he" data-title="חוזק חומרים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%A1%E1%83%90%E1%83%9A%E1%83%90%E1%83%97%E1%83%90_%E1%83%92%E1%83%90%E1%83%9B%E1%83%AB%E1%83%9A%E1%83%94%E1%83%9D%E1%83%91%E1%83%90" title="მასალათა გამძლეობა – Georgian" lang="ka" hreflang="ka" data-title="მასალათა გამძლეობა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D1%80%D0%B8%D0%B0%D0%BB%D0%B4%D0%B0%D1%80_%D0%BA%D0%B5%D0%B4%D0%B5%D1%80%D0%B3%D1%96%D1%81%D1%96" title="Материалдар кедергісі – Kazakh" lang="kk" hreflang="kk" data-title="Материалдар кедергісі" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Materi%C4%81lu_pretest%C4%ABba" title="Materiālu pretestība – Latvian" lang="lv" hreflang="lv" data-title="Materiālu pretestība" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B4%AC%E0%B4%B2%E0%B4%A4" title="പ്രബലത – Malayalam" lang="ml" hreflang="ml" data-title="പ്രബലത" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kekuatan_bahan" title="Kekuatan bahan – Malay" lang="ms" hreflang="ms" data-title="Kekuatan bahan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%91%D0%B0%D1%82_%D0%B1%D3%A9%D1%85%D0%B8%D0%B9%D0%BD_%D0%BE%D0%BD%D0%BE%D0%BB" title="Бат бөхийн онол – Mongolian" lang="mn" hreflang="mn" data-title="Бат бөхийн онол" 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/Sterkteleer" title="Sterkteleer – Dutch" lang="nl" hreflang="nl" data-title="Sterkteleer" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%9D%90%E6%96%99%E5%8A%9B%E5%AD%A6" title="材料力学 – Japanese" lang="ja" hreflang="ja" data-title="材料力学" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Fasthetsl%C3%A6re" title="Fasthetslære – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Fasthetslære" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Fastleiksl%C3%A6re" title="Fastleikslære – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Fastleikslære" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wytrzyma%C5%82o%C5%9B%C4%87_materia%C5%82%C3%B3w" title="Wytrzymałość materiałów – Polish" lang="pl" hreflang="pl" data-title="Wytrzymałość materiałów" data-language-autonym="Polski" 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data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Behavior of solid objects subject to stresses and strains</div> <p> The field of <b>strength of materials</b> (also called <b>mechanics of materials</b>) typically refers to various methods of calculating the <a href="/wiki/Stress_(physics)" class="mw-redirect" title="Stress (physics)">stresses</a> and <a href="/wiki/Strain_(physics)" class="mw-redirect" title="Strain (physics)">strains</a> in structural members, such as beams, columns, and shafts. The methods employed to predict the response of a structure under loading and its susceptibility to various failure modes takes into account the properties of the materials such as its <a href="/wiki/Yield_strength" class="mw-redirect" title="Yield strength">yield strength</a>, <a href="/wiki/Ultimate_strength" class="mw-redirect" title="Ultimate strength">ultimate strength</a>, <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a>, and <a href="/wiki/Poisson%27s_ratio" title="Poisson's ratio">Poisson's ratio</a>. In addition, the mechanical element's macroscopic properties (geometric properties) such as its length, width, thickness, boundary constraints and abrupt changes in geometry such as holes are considered. </p><p>The theory began with the consideration of the behavior of one and two dimensional members of structures, whose states of stress can be approximated as two dimensional, and was then generalized to three dimensions to develop a more complete theory of the elastic and plastic behavior of materials. An important founding pioneer in mechanics of materials was <a href="/wiki/Stephen_Timoshenko" title="Stephen Timoshenko">Stephen Timoshenko</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strength_of_materials&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the mechanics of materials, the strength of a material is its ability to withstand an applied load without failure or <a href="/wiki/Deformation_(engineering)#Plastic_deformation" title="Deformation (engineering)">plastic deformation</a>. The field of strength of materials deals with forces and deformations that result from their acting on a material. A load applied to a mechanical member will induce internal forces within the member called stresses when those forces are expressed on a unit basis. The stresses acting on the material cause deformation of the material in various manners including breaking them completely. Deformation of the material is called strain when those deformations too are placed on a unit basis. </p><p>The stresses and strains that develop within a mechanical member must be calculated in order to assess the load capacity of that member. This requires a complete description of the geometry of the member, its constraints, the loads applied to the member and the properties of the material of which the member is composed. The applied loads may be axial (tensile or compressive), or rotational (strength shear). With a complete description of the loading and the geometry of the member, the state of stress and state of strain at any point within the member can be calculated. Once the state of stress and strain within the member is known, the strength (load carrying capacity) of that member, its deformations (stiffness qualities), and its stability (ability to maintain its original configuration) can be calculated. </p><p>The calculated stresses may then be compared to some measure of the strength of the member such as its material yield or ultimate strength. The calculated deflection of the member may be compared to deflection criteria that are based on the member's use. The calculated buckling load of the member may be compared to the applied load. The calculated stiffness and mass distribution of the member may be used to calculate the member's dynamic response and then compared to the acoustic environment in which it will be used. </p><p>Material strength refers to the point on the engineering <a href="/wiki/Stress%E2%80%93strain_curve" title="Stress–strain curve">stress–strain curve</a> (yield stress) beyond which the material experiences deformations that will not be completely reversed upon removal of the loading and as a result, the member will have a permanent deflection. The ultimate strength of the material refers to the maximum value of stress reached. The fracture strength is the stress value at fracture (the last stress value recorded). </p> <div class="mw-heading mw-heading3"><h3 id="Types_of_loadings">Types of loadings</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strength_of_materials&action=edit&section=2" title="Edit section: Types of loadings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Transverse_plane" title="Transverse plane">Transverse</a> loadings – Forces applied perpendicular to the longitudinal axis of a member. Transverse loading causes the member to bend and deflect from its original position, with internal tensile and compressive strains accompanying the change in curvature of the member.<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> Transverse loading also induces shear forces that cause shear deformation of the material and increase the transverse deflection of the member.</li> <li>Axial loading – The applied forces are collinear with the longitudinal axis of the member. The forces cause the member to either stretch or shorten.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Torsion_(mechanics)" title="Torsion (mechanics)">Torsional</a> loading – Twisting action caused by a pair of externally applied equal and oppositely directed force couples acting on parallel planes or by a single external couple applied to a member that has one end fixed against rotation.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Stress_terms">Stress terms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strength_of_materials&action=edit&section=3" title="Edit section: Stress terms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Compressive_tensile_shear_loading.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Compressive_tensile_shear_loading.svg/220px-Compressive_tensile_shear_loading.svg.png" decoding="async" width="220" height="141" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Compressive_tensile_shear_loading.svg/330px-Compressive_tensile_shear_loading.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/Compressive_tensile_shear_loading.svg/440px-Compressive_tensile_shear_loading.svg.png 2x" data-file-width="360" data-file-height="230" /></a><figcaption>A material being loaded in a) compression, b) tension, c) shear</figcaption></figure> <p>Uniaxial stress is expressed by </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 \sigma ={\frac {F}{A}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>F</mi> <mi>A</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ={\frac {F}{A}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10cf8f2f4ed3dbb09e09393fda6bc1ed14360b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.654ex; height:5.343ex;" alt="{\displaystyle \sigma ={\frac {F}{A}},}"></span></dd></dl> <p>where <i>F</i> is the force acting on an area <i>A</i>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The area can be the undeformed area or the deformed area, depending on whether <a href="/wiki/Engineering_stress#Stress_in_one-dimensional_bodies" class="mw-redirect" title="Engineering stress">engineering stress</a> or true stress is of interest. </p> <ul><li><i><a href="/wiki/Compressive_stress" title="Compressive stress">Compressive stress</a></i> (or <a href="/wiki/Compression_(physics)" title="Compression (physics)">compression</a>) is the stress state caused by an applied load that acts to reduce the length of the material (<a href="/wiki/Compression_member" title="Compression member">compression member</a>) along the axis of the applied load; it is, in other words, a stress state that causes a squeezing of the material. A simple case of compression is the uniaxial compression induced by the action of opposite, pushing forces. Compressive strength for materials is generally higher than their tensile strength. However, structures loaded in compression are subject to additional failure modes, such as <a href="/wiki/Buckling" title="Buckling">buckling</a>, that are dependent on the member's geometry.</li> <li><i><a href="/wiki/Tensile_stress" class="mw-redirect" title="Tensile stress">Tensile stress</a></i> is the stress state caused by an applied load that tends to elongate the material along the axis of the applied load, in other words, the stress caused by <i>pulling</i> the material. The strength of structures of equal cross-sectional area loaded in tension is independent of shape of the cross-section. Materials loaded in tension are susceptible to <a href="/wiki/Stress_concentration" title="Stress concentration">stress concentrations</a> such as material defects or abrupt changes in geometry. However, materials exhibiting ductile behaviour (many metals for example) can tolerate some defects while brittle materials (such as ceramics and some steels) can fail well below their ultimate material strength.</li> <li><i><a href="/wiki/Shear_stress" title="Shear stress">Shear stress</a></i> is the stress state caused by the combined energy of a pair of opposing forces acting along parallel lines of action through the material, in other words, the stress caused by faces of the material <i>sliding</i> relative to one another. An example is cutting paper with <a href="/wiki/Scissor" class="mw-redirect" title="Scissor">scissors</a><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> or stresses due to torsional loading.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Stress_parameters_for_resistance">Stress parameters for resistance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strength_of_materials&action=edit&section=4" title="Edit section: Stress parameters for resistance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Material resistance can be expressed in several <a href="/wiki/Mechanical_stress" class="mw-redirect" title="Mechanical stress">mechanical stress</a> parameters. The term <i>material strength</i> is used when referring to <i>mechanical stress</i> parameters. These are <a href="/wiki/Physical_quantity" title="Physical quantity">physical quantities</a> with dimension homogeneous to <i>pressure</i> and <i>force per unit surface</i>. The traditional measure unit for strength are therefore <a href="/wiki/Megapascal" class="mw-redirect" title="Megapascal">MPa</a> in the <a href="/wiki/International_System_of_Units" title="International System of Units">International System of Units</a>, and the <a href="/wiki/Pounds_per_square_inch" class="mw-redirect" title="Pounds per square inch">psi</a> between the <a href="/wiki/United_States_customary_units" title="United States customary units">United States customary units</a>. Strength parameters include: yield strength, tensile strength, fatigue strength, crack resistance, and other parameters.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2020)">citation needed</span></a></i>]</sup> </p> <ul><li><i><a href="/wiki/Yield_(engineering)" title="Yield (engineering)">Yield strength</a></i> is the lowest stress that produces a permanent deformation in a material. In some materials, like <a href="/wiki/Aluminium_alloy" title="Aluminium alloy">aluminium alloys</a>, the point of yielding is difficult to identify, thus it is usually defined as the stress required to cause 0.2% plastic strain. This is called a 0.2% proof stress.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul> <ul><li><i><a href="/wiki/Compressive_strength" title="Compressive strength">Compressive strength</a></i> is a limit state of <a href="/wiki/Compressive_stress" title="Compressive stress">compressive stress</a> that leads to failure in a material in the manner of ductile failure (infinite theoretical yield) or brittle failure (rupture as the result of crack propagation, or sliding along a weak plane – see <a href="/wiki/Shear_strength" title="Shear strength">shear strength</a>).</li> <li><i><a href="/wiki/Tensile_strength" class="mw-redirect" title="Tensile strength">Tensile strength</a></i> or <i>ultimate tensile strength</i> is a limit state of <a href="/wiki/Tensile_stress" class="mw-redirect" title="Tensile stress">tensile stress</a> that leads to tensile failure in the manner of ductile failure (yield as the first stage of that failure, some hardening in the second stage and breakage after a possible "neck" formation) or brittle failure (sudden breaking in two or more pieces at a low-stress state). The tensile strength can be quoted as either true stress or engineering stress, but engineering stress is the most commonly used.</li> <li><i><a href="/wiki/Fatigue_(material)" title="Fatigue (material)">Fatigue strength</a></i> is a more complex measure of the strength of a material that considers several loading episodes in the service period of an object,<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> and is usually more difficult to assess than the static strength measures. Fatigue strength is quoted here as a simple <a href="/wiki/Range_(statistics)" title="Range (statistics)">range</a> (<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 \Delta \sigma =\sigma _{\mathrm {max} }-\sigma _{\mathrm {min} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>σ</mi> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \sigma =\sigma _{\mathrm {max} }-\sigma _{\mathrm {min} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebbec91345be2fc73ed6e680b9753833b3ac6af6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.123ex; height:2.509ex;" alt="{\displaystyle \Delta \sigma =\sigma _{\mathrm {max} }-\sigma _{\mathrm {min} }}"></span>). In the case of <a href="/wiki/Cyclic_loading" class="mw-redirect" title="Cyclic loading">cyclic loading</a> it can be appropriately expressed as an <a href="/wiki/Amplitude" title="Amplitude">amplitude</a> usually at zero mean stress, along with the number of cycles to failure under that condition of stress.</li></ul> <ul><li><i><a href="/wiki/Impact_strength" class="mw-redirect" title="Impact strength">Impact strength</a></i> is the capability of the material to withstand a suddenly applied load and is expressed in terms of energy. Often measured with the <a href="/wiki/Izod_impact_strength_test" title="Izod impact strength test">Izod impact strength test</a> or <a href="/wiki/Charpy_impact_test" title="Charpy impact test">Charpy impact test</a>, both of which measure the impact energy required to fracture a sample. Volume, modulus of <a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">elasticity</a>, distribution of forces, and yield strength affect the impact strength of a material. In order for a material or object to have a high impact strength, the stresses must be distributed evenly throughout the object. It also must have a large volume with a low modulus of elasticity and a high material yield strength.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Strain_parameters_for_resistance">Strain parameters for resistance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strength_of_materials&action=edit&section=5" title="Edit section: Strain parameters for resistance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><i><a href="/wiki/Deformation_(engineering)" title="Deformation (engineering)">Deformation</a></i> of the material is the change in geometry created when stress is applied (as a result of applied forces, gravitational fields, accelerations, thermal expansion, etc.). Deformation is expressed by the displacement field of the material.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li> <li><i><a href="/wiki/Strain_(materials_science)" class="mw-redirect" title="Strain (materials science)">Strain</a></i>, or <i>reduced deformation</i>, is a mathematical term that expresses the trend of the deformation change among the material field. Strain is the deformation per unit length.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> In the case of uniaxial loading the displacement of a specimen (for example, a bar element) lead to a calculation of strain expressed as the quotient of the displacement and the original length of the specimen. For 3D displacement fields it is expressed as derivatives of displacement functions in terms of a second-order <a href="/wiki/Tensor" title="Tensor">tensor</a> (with 6 independent elements).</li> <li><i><a href="/wiki/Deflection_(engineering)" title="Deflection (engineering)">Deflection</a></i> is a term to describe the magnitude to which a structural element is displaced when subject to an applied load.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading3"><h3 id="Stress–strain_relations"><span id="Stress.E2.80.93strain_relations"></span>Stress–strain relations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strength_of_materials&action=edit&section=6" title="Edit section: Stress–strain relations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Stress%E2%80%93strain_curve" title="Stress–strain curve">Stress–strain curve</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Tension_test.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Tension_test.svg/300px-Tension_test.svg.png" decoding="async" width="300" height="232" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Tension_test.svg/450px-Tension_test.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/Tension_test.svg/600px-Tension_test.svg.png 2x" data-file-width="502" data-file-height="388" /></a><figcaption>Basic static response of a specimen under tension</figcaption></figure> <ul><li><i><a href="/wiki/Elasticity_(physics)" title="Elasticity (physics)">Elasticity</a></i> is the ability of a material to return to its previous shape after stress is released. In many materials, the relation between applied stress is directly proportional to the resulting strain (up to a certain limit), and a graph representing those two quantities is a straight line.</li></ul> <p>The slope of this line is known as <a href="/wiki/Young%27s_modulus" title="Young's modulus">Young's modulus</a>, or the "modulus of elasticity". The modulus of elasticity can be used to determine the stress–strain relationship in the linear-elastic portion of the stress–strain curve. The linear-elastic region is either below the yield point, or if a yield point is not easily identified on the stress–strain plot it is defined to be between 0 and 0.2% strain, and is defined as the region of strain in which no yielding (permanent deformation) occurs.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <ul><li><i><a href="/wiki/Plasticity_(physics)" title="Plasticity (physics)">Plasticity</a></i> or plastic deformation is the opposite of elastic deformation and is defined as unrecoverable strain. Plastic deformation is retained after the release of the applied stress. Most materials in the linear-elastic category are usually capable of plastic deformation. Brittle materials, like ceramics, do not experience any plastic deformation and will fracture under relatively low strain, while ductile materials such as metallics, lead, or polymers will plastically deform much more before a fracture initiation.</li></ul> <p>Consider the difference between a carrot and chewed bubble gum. The carrot will stretch very little before breaking. The chewed bubble gum, on the other hand, will plastically deform enormously before finally breaking. </p> <div class="mw-heading mw-heading2"><h2 id="Design_terms">Design terms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strength_of_materials&action=edit&section=7" title="Edit section: Design terms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ultimate strength is an attribute related to a material, rather than just a specific specimen made of the material, and as such it is quoted as the force per unit of cross section area (N/m<sup>2</sup>). The ultimate strength is the maximum stress that a material can withstand before it breaks or weakens.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> For example, the ultimate tensile strength (UTS) of AISI 1018 Steel is 440 <a href="/wiki/Megapascal" class="mw-redirect" title="Megapascal">MPa</a>. In Imperial units, the unit of stress is given as lbf/in<sup>2</sup> or <a href="/wiki/Pounds-force_per_square_inch" class="mw-redirect" title="Pounds-force per square inch">pounds-force per square inch</a>. This unit is often abbreviated as <b>psi</b>. One thousand psi is abbreviated <b>ksi</b>. </p><p>A <a href="/wiki/Factor_of_safety" title="Factor of safety">factor of safety</a> is a design criteria that an engineered component or structure must achieve. <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 FS=F/f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mi>S</mi> <mo>=</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle FS=F/f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b916dab6b2487845a2a970b3c1631f111d0e403" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.52ex; height:2.843ex;" alt="{\displaystyle FS=F/f}"></span>, where FS: the factor of safety, Rf The applied stress, and F: ultimate allowable stress (psi or MPa)<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>Margin of Safety is the common method for design criteria. It is defined MS = P<sub>u</sub>/P − 1. </p><p>For example, to achieve a factor of safety of 4, the allowable stress in an AISI 1018 steel component can be calculated to be <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 F=UTS/FS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>U</mi> <mi>T</mi> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>F</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=UTS/FS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa2bdc16875de3dca5fa332570623d49d7869184" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.16ex; height:2.843ex;" alt="{\displaystyle F=UTS/FS}"></span> = 440/4 = 110 MPa, or <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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> = 110×10<sup>6</sup> N/m<sup>2</sup>. Such allowable stresses are also known as "design stresses" or "working stresses". </p><p>Design stresses that have been determined from the ultimate or yield point values of the materials give safe and reliable results only for the case of static loading. Many machine parts fail when subjected to a non-steady and continuously varying loads even though the developed stresses are below the yield point. Such failures are called fatigue failure. The failure is by a fracture that appears to be brittle with little or no visible evidence of yielding. However, when the stress is kept below "fatigue stress" or "endurance limit stress", the part will endure indefinitely. A purely reversing or cyclic stress is one that alternates between equal positive and negative peak stresses during each cycle of operation. In a purely cyclic stress, the average stress is zero. When a part is subjected to a cyclic stress, also known as stress range (Sr), it has been observed that the failure of the part occurs after a number of stress reversals (N) even if the magnitude of the stress range is below the material's yield strength. Generally, higher the range stress, the fewer the number of reversals needed for failure. </p> <div class="mw-heading mw-heading3"><h3 id="Failure_theories">Failure theories</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strength_of_materials&action=edit&section=8" title="Edit section: Failure theories"><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/Material_failure_theory" title="Material failure theory">Material failure theory</a></div> <p>There are four failure theories: maximum shear stress theory, maximum normal stress theory, maximum strain energy theory, and maximum distortion energy theory (von Mises criterion of failure). Out of these four theories of failure, the maximum normal stress theory is only applicable for brittle materials, and the remaining three theories are applicable for ductile materials. Of the latter three, the distortion energy theory provides the most accurate results in a majority of the stress conditions. The strain energy theory needs the value of <a href="/wiki/Poisson%27s_ratio" title="Poisson's ratio">Poisson's ratio</a> of the part material, which is often not readily available. The maximum shear stress theory is conservative. For simple unidirectional normal stresses all theories are equivalent, which means all theories will give the same result. </p> <ul><li><b>Maximum shear stress theory</b> postulates that failure will occur if the magnitude of the maximum shear stress in the part exceeds the shear strength of the material determined from uniaxial testing.</li> <li><b>Maximum normal stress theory</b> postulates that failure will occur if the maximum normal stress in the part exceeds the ultimate tensile stress of the material as determined from uniaxial testing. This theory deals with brittle materials only. The maximum tensile stress should be less than or equal to ultimate tensile stress divided by factor of safety. The magnitude of the maximum compressive stress should be less than ultimate compressive stress divided by factor of safety.</li> <li><b>Maximum strain energy theory</b> postulates that failure will occur when the strain energy per unit volume due to the applied stresses in a part equals the strain energy per unit volume at the yield point in uniaxial testing.</li> <li><b>Maximum distortion energy theory</b>, also known as maximum distortion energy theory of failure or <a href="/wiki/Von_Mises_yield_criterion" title="Von Mises yield criterion">von Mises–Hencky theory</a>. This theory postulates that failure will occur when the distortion energy per unit volume due to the applied stresses in a part equals the distortion energy per unit volume at the yield point in uniaxial testing. The total elastic energy due to strain can be divided into two parts: one part causes change in volume, and the other part causes a change in shape. Distortion energy is the amount of energy that is needed to change the shape.</li> <li>Fracture mechanics was established by <a href="/wiki/Alan_Arnold_Griffith" title="Alan Arnold Griffith">Alan Arnold Griffith</a> and <a href="/wiki/George_Rankine_Irwin" title="George Rankine Irwin">George Rankine Irwin</a>. This important theory is also known as numeric conversion of toughness of material in the case of crack existence.</li></ul> <p>A material's strength is dependent on its <a href="/wiki/Microstructure" title="Microstructure">microstructure</a>. The engineering processes to which a material is subjected can alter this microstructure. The variety of <a href="/wiki/Strengthening_mechanisms_of_materials" title="Strengthening mechanisms of materials">strengthening mechanisms</a> that alter the strength of a material includes <a href="/wiki/Work_hardening" title="Work hardening">work hardening</a>, <a href="/wiki/Solid_solution_strengthening" title="Solid solution strengthening">solid solution strengthening</a>, <a href="/wiki/Precipitation_hardening" title="Precipitation hardening">precipitation hardening</a>, and <a href="/wiki/Grain_boundary_strengthening" title="Grain boundary strengthening">grain boundary strengthening</a> and can be quantitatively and qualitatively explained. Strengthening mechanisms are accompanied by the caveat that some other mechanical properties of the material may degenerate in an attempt to make the material stronger. For example, in grain boundary strengthening, although <a href="/wiki/Yield_strength" class="mw-redirect" title="Yield strength">yield strength</a> is maximized with decreasing grain size, ultimately, very small grain sizes make the material brittle. In general, the yield strength of a material is an adequate indicator of the material's mechanical strength. Considered in tandem with the fact that the yield strength is the parameter that predicts <a href="/wiki/Plastic_deformation" class="mw-redirect" title="Plastic deformation">plastic deformation</a> in the material, one can make informed decisions on how to increase the strength of a material depending on its microstructural properties and the desired end effect. Strength is expressed in terms of the limiting values of the <a href="/wiki/Compressive_stress" title="Compressive stress">compressive stress</a>, <a href="/wiki/Tensile_stress" class="mw-redirect" title="Tensile stress">tensile stress</a>, and <a href="/wiki/Shear_stress" title="Shear stress">shear stresses</a> that would cause failure. The effects of dynamic loading are probably the most important practical consideration of the theory of elasticity, especially the problem of <a href="/wiki/Fatigue_(material)" title="Fatigue (material)">fatigue</a>. Repeated loading often initiates cracks, which grow until failure occurs at the corresponding residual strength of the structure. Cracks always start at a <a href="/wiki/Stress_concentration" title="Stress concentration">stress concentrations</a> especially changes in cross-section of the product or defects in manufacturing, near holes and corners at nominal stress levels far lower than those quoted for the strength of the material. </p> <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=Strength_of_materials&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Creep_(deformation)" title="Creep (deformation)">Creep (deformation)</a> – Tendency of a solid material to move slowly or deform permanently under mechanical stress</li> <li><a href="/wiki/Deformation_mechanism_map" class="mw-redirect" title="Deformation mechanism map">Deformation mechanism map</a> – Microscopic processes responsible for changes in a material's structure, shape and volume<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Dynamics_(mechanics)" class="mw-redirect" title="Dynamics (mechanics)">Dynamics</a> – Branch of mechanics studying forces and their effect on motion</li> <li><a href="/wiki/Fatigue_(material)" title="Fatigue (material)">Fatigue (material)</a> – Initiation and propagation of cracks in a material due to cyclic loading</li> <li><a href="/wiki/Forensic_engineering" title="Forensic engineering">Forensic engineering</a> – Investigation of failures associated with legal intervention</li> <li><a href="/wiki/Fracture_mechanics" title="Fracture mechanics">Fracture mechanics</a> – Study of propagation of cracks in materials</li> <li><a href="/wiki/Fracture_toughness" title="Fracture toughness">Fracture toughness</a> – Stress intensity factor at which a crack's propagation increases drastically</li> <li><a href="/wiki/List_of_materials_properties#Mechanical_properties" title="List of materials properties">List of materials properties § Mechanical properties</a></li> <li><a href="/wiki/Material_selection" title="Material selection">Material selection</a> – Step in the process of designing physical objects</li> <li><a href="/wiki/Molecular_diffusion" title="Molecular diffusion">Molecular diffusion</a> – Thermal motion of liquid or gas particles at temperatures above absolute zero</li> <li><a href="/wiki/Specific_strength" title="Specific strength">Specific strength</a> – Ratio of strength to mass for a material</li> <li><a href="/wiki/Statics" title="Statics">Statics</a> – Branch of mechanics concerned with balance of forces in nonmoving systems</li> <li><a href="/wiki/Universal_testing_machine" title="Universal testing machine">Universal testing machine</a> – Type of equipment for determining tensile or compressive strength of a material</li></ul> </div> <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=Strength_of_materials&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBeer_&_Johnston2006" class="citation book cs1">Beer & Johnston (2006). <i>Mechanics of Materials</i> (5th ed.). McGraw Hill. p. 210. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-352938-7" title="Special:BookSources/978-0-07-352938-7"><bdi>978-0-07-352938-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+Materials&rft.pages=210&rft.edition=5th&rft.pub=McGraw+Hill&rft.date=2006&rft.isbn=978-0-07-352938-7&rft.au=Beer+%26+Johnston&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrength+of+materials" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeer_&_Johnston2006" class="citation book cs1">Beer & Johnston (2006). <i>Mechanics of Materials</i> (5th ed.). McGraw Hill. p. 7. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-352938-7" title="Special:BookSources/978-0-07-352938-7"><bdi>978-0-07-352938-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+Materials&rft.pages=7&rft.edition=5th&rft.pub=McGraw+Hill&rft.date=2006&rft.isbn=978-0-07-352938-7&rft.au=Beer+%26+Johnston&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrength+of+materials" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeer_&_Johnston2006" class="citation book cs1">Beer & Johnston (2006). <i>Mechanics of Materials</i> (5th ed.). McGraw Hill. p. 5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-352938-7" title="Special:BookSources/978-0-07-352938-7"><bdi>978-0-07-352938-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+Materials&rft.pages=5&rft.edition=5th&rft.pub=McGraw+Hill&rft.date=2006&rft.isbn=978-0-07-352938-7&rft.au=Beer+%26+Johnston&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrength+of+materials" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeer_&_Johnston2006" class="citation book cs1">Beer & Johnston (2006). <i>Mechanics of Materials</i> (5th ed.). McGraw Hill. pp. 9–10. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-352938-7" title="Special:BookSources/978-0-07-352938-7"><bdi>978-0-07-352938-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+Materials&rft.pages=9-10&rft.edition=5th&rft.pub=McGraw+Hill&rft.date=2006&rft.isbn=978-0-07-352938-7&rft.au=Beer+%26+Johnston&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrength+of+materials" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeerJohnstonDewolf2009" class="citation book cs1">Beer, Ferdinand Pierre; Johnston, Elwood Russell; Dewolf, John T (2009). <i>Mechanics of Materials</i> (5th ed.). p. 52. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-352938-7" title="Special:BookSources/978-0-07-352938-7"><bdi>978-0-07-352938-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+Materials&rft.pages=52&rft.edition=5th&rft.date=2009&rft.isbn=978-0-07-352938-7&rft.aulast=Beer&rft.aufirst=Ferdinand+Pierre&rft.au=Johnston%2C+Elwood+Russell&rft.au=Dewolf%2C+John+T&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrength+of+materials" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeer_&_Johnston2006" class="citation book cs1">Beer & Johnston (2006). <i>Mechanics of Materials</i> (5th ed.). McGraw Hill. p. 60. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-352938-7" title="Special:BookSources/978-0-07-352938-7"><bdi>978-0-07-352938-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+Materials&rft.pages=60&rft.edition=5th&rft.pub=McGraw+Hill&rft.date=2006&rft.isbn=978-0-07-352938-7&rft.au=Beer+%26+Johnston&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrength+of+materials" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeer_&_Johnston2006" class="citation book cs1">Beer & Johnston (2006). <i>Mechanics of Materials</i> (5th ed.). McGraw Hill. pp. 693–696. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-352938-7" title="Special:BookSources/978-0-07-352938-7"><bdi>978-0-07-352938-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+Materials&rft.pages=693-696&rft.edition=5th&rft.pub=McGraw+Hill&rft.date=2006&rft.isbn=978-0-07-352938-7&rft.au=Beer+%26+Johnston&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrength+of+materials" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeer_&_Johnston2006" class="citation book cs1">Beer & Johnston (2006). <i>Mechanics of Materials</i> (5th ed.). McGraw Hill. p. 47. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-352938-7" title="Special:BookSources/978-0-07-352938-7"><bdi>978-0-07-352938-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+Materials&rft.pages=47&rft.edition=5th&rft.pub=McGraw+Hill&rft.date=2006&rft.isbn=978-0-07-352938-7&rft.au=Beer+%26+Johnston&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrength+of+materials" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeer_&_Johnston2006" class="citation book cs1">Beer & Johnston (2006). <i>Mechanics of Materials</i> (5th ed.). McGraw Hill. p. 49. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-352938-7" title="Special:BookSources/978-0-07-352938-7"><bdi>978-0-07-352938-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+Materials&rft.pages=49&rft.edition=5th&rft.pub=McGraw+Hill&rft.date=2006&rft.isbn=978-0-07-352938-7&rft.au=Beer+%26+Johnston&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrength+of+materials" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFR._C._Hibbeler2009" class="citation book cs1">R. C. Hibbeler (2009). <i>Structural Analysis</i> (7 ed.). Pearson Prentice Hall. p. 305. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-602060-8" title="Special:BookSources/978-0-13-602060-8"><bdi>978-0-13-602060-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Structural+Analysis&rft.pages=305&rft.edition=7&rft.pub=Pearson+Prentice+Hall&rft.date=2009&rft.isbn=978-0-13-602060-8&rft.au=R.+C.+Hibbeler&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrength+of+materials" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeer_&_Johnston2006" class="citation book cs1">Beer & Johnston (2006). <i>Mechanics of Materials</i> (5th ed.). McGraw Hill. pp. 53–56. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-352938-7" title="Special:BookSources/978-0-07-352938-7"><bdi>978-0-07-352938-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+Materials&rft.pages=53-56&rft.edition=5th&rft.pub=McGraw+Hill&rft.date=2006&rft.isbn=978-0-07-352938-7&rft.au=Beer+%26+Johnston&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrength+of+materials" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeer_&_Johnston2006" class="citation book cs1">Beer & Johnston (2006). <i>Mechanics of Materials</i> (5thv ed.). McGraw Hill. pp. 27–28. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-352938-7" title="Special:BookSources/978-0-07-352938-7"><bdi>978-0-07-352938-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+Materials&rft.pages=27-28&rft.edition=5thv&rft.pub=McGraw+Hill&rft.date=2006&rft.isbn=978-0-07-352938-7&rft.au=Beer+%26+Johnston&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrength+of+materials" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeer_&_Johnston2006" class="citation book cs1">Beer & Johnston (2006). <i>Mechanics of Materials</i> (5th ed.). McGraw Hill. p. 28. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-352938-7" title="Special:BookSources/978-0-07-352938-7"><bdi>978-0-07-352938-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+Materials&rft.pages=28&rft.edition=5th&rft.pub=McGraw+Hill&rft.date=2006&rft.isbn=978-0-07-352938-7&rft.au=Beer+%26+Johnston&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStrength+of+materials" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strength_of_materials&action=edit&section=11" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col"> <ul><li>Fa-Hwa Cheng, Initials. (1997). Strength of material. Ohio: McGraw-Hill</li> <li>Mechanics of Materials, E.J. Hearn</li> <li>Alfirević, Ivo. <i>Strength of Materials I</i>. Tehnička knjiga, 1995. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/953-172-010-X" title="Special:BookSources/953-172-010-X">953-172-010-X</a>.</li> <li>Alfirević, Ivo. <i>Strength of Materials II</i>. Tehnička knjiga, 1999. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/953-6168-85-5" title="Special:BookSources/953-6168-85-5">953-6168-85-5</a>.</li> <li><a href="/wiki/M._F._Ashby" class="mw-redirect" title="M. F. Ashby">Ashby, M.F.</a> <i>Materials Selection in Design</i>. Pergamon, 1992.</li> <li>Beer, F.P., E.R. Johnston, et al. <i>Mechanics of Materials</i>, 3rd edition. McGraw-Hill, 2001. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-248673-2" title="Special:BookSources/0-07-248673-2">0-07-248673-2</a></li> <li>Cottrell, A.H. <i>Mechanical Properties of Matter</i>. Wiley, New York, 1964.</li> <li>Den Hartog, Jacob P. <i>Strength of Materials</i>. Dover Publications, Inc., 1961, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-60755-0" title="Special:BookSources/0-486-60755-0">0-486-60755-0</a>.</li> <li>Drucker, D.C. <i>Introduction to Mechanics of Deformable Solids</i>. McGraw-Hill, 1967.</li> <li><a href="/wiki/J.E._Gordon" class="mw-redirect" title="J.E. Gordon">Gordon, J.E.</a> <i>The New Science of Strong Materials</i>. Princeton, 1984.</li> <li>Groover, Mikell P. <i>Fundamentals of Modern Manufacturing</i>, 2nd edition. John Wiley & Sons, Inc., 2002. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-40051-3" title="Special:BookSources/0-471-40051-3">0-471-40051-3</a>.</li> <li>Hashemi, Javad and William F. Smith. <i>Foundations of Materials Science and Engineering</i>, 4th edition. McGraw-Hill, 2006. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-125690-3" title="Special:BookSources/0-07-125690-3">0-07-125690-3</a>.</li> <li>Hibbeler, R.C. <i>Statics and Mechanics of Materials</i>, SI Edition. Prentice-Hall, 2004. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-129011-8" title="Special:BookSources/0-13-129011-8">0-13-129011-8</a>.</li> <li>Lebedev, Leonid P. and Michael J. Cloud. <i>Approximating Perfection: A Mathematician's Journey into the World of Mechanics</i>. Princeton University Press, 2004. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-11726-8" title="Special:BookSources/0-691-11726-8">0-691-11726-8</a>.</li> <li><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016/B978-0-12-394584-6.00010-8">Chapter 10 – Strength of Elastomers</a>, A.N. Gent, W.V. Mars, In: James E. Mark, Burak Erman and Mike Roland, Editor(s), The Science and Technology of Rubber (Fourth Edition), Academic Press, Boston, 2013, Pages 473–516, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780123945846" title="Special:BookSources/9780123945846">9780123945846</a>, 10.1016/B978-0-12-394584-6.00010-8</li> <li>Mott, Robert L. <i>Applied Strength of Materials</i>, 4th edition. Prentice-Hall, 2002. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-088578-9" title="Special:BookSources/0-13-088578-9">0-13-088578-9</a>.</li> <li>Popov, Egor P. <i>Engineering Mechanics of Solids</i>. Prentice Hall, Englewood Cliffs, N. J., 1990. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-279258-3" title="Special:BookSources/0-13-279258-3">0-13-279258-3</a>.</li> <li>Ramamrutham, S. <i>Strength of Materials</i>.</li> <li>Shames, I.H. and F.A. Cozzarelli. <i>Elastic and inelastic stress analysis</i>. Prentice-Hall, 1991. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-56032-686-7" title="Special:BookSources/1-56032-686-7">1-56032-686-7</a>.</li> <li><a href="/wiki/Stephen_Timoshenko" title="Stephen Timoshenko">Timoshenko S.</a> <i>Strength of Materials</i>, 3rd edition. Krieger Publishing Company, 1976, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-88275-420-3" title="Special:BookSources/0-88275-420-3">0-88275-420-3</a>.</li> <li>Timoshenko, S.P. and D.H. Young. <i>Elements of Strength of Materials</i>, 5th edition. (MKS System)</li> <li>Davidge, R.W., Mechanical Behavior of Ceramics, Cambridge Solid State Science Series, (1979)</li> <li>Lawn, B.R., Fracture of Brittle Solids, Cambridge Solid State Science Series, 2nd Edn. (1993)</li> <li>Green, D., An Introduction to the Mechanical Properties of Ceramics, Cambridge Solid State Science Series, Eds. Clarke, D.R., Suresh, S., Ward, I.M.Babu Tom.K (1998)</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Strength_of_materials&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.mech.uwa.edu.au/DANotes/SSS/failure/theories.html">Failure theories</a></li> <li><a rel="nofollow" class="external text" href="http://materials.open.ac.uk/mem/index.htm">Case studies in structural failure</a></li></ul> <div class="navbox-styles"><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 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