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href="https://bg.wikipedia.org/wiki/%D0%A3%D1%81%D1%83%D0%BA%D0%B2%D0%B0%D0%BD%D0%B5" title="Усукване – Bulgarian" lang="bg" hreflang="bg" data-title="Усукване" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Torzija_(mehanika)" title="Torzija (mehanika) – Bosnian" lang="bs" hreflang="bs" data-title="Torzija (mehanika)" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Torsi%C3%B3_(mec%C3%A0nica)" title="Torsió (mecànica) – Catalan" lang="ca" hreflang="ca" data-title="Torsió (mecànica)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Krut" title="Krut – Czech" lang="cs" hreflang="cs" data-title="Krut" 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/Torsion_(Mechanik)" title="Torsion (Mechanik) – German" lang="de" hreflang="de" data-title="Torsion (Mechanik)" 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/V%C3%A4%C3%A4ne" title="Vääne – Estonian" lang="et" hreflang="et" data-title="Vääne" 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%A3%CF%84%CF%81%CE%AD%CF%88%CE%B7" title="Στρέψη – Greek" lang="el" hreflang="el" data-title="Στρέψη" 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/Torsi%C3%B3n_mec%C3%A1nica" title="Torsión mecánica – Spanish" lang="es" hreflang="es" data-title="Torsión mecánica" 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/Tordo" title="Tordo – Esperanto" lang="eo" hreflang="eo" data-title="Tordo" 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/Bihurdura_(mekanika)" title="Bihurdura (mekanika) – Basque" lang="eu" hreflang="eu" data-title="Bihurdura (mekanika)" 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%BE%DB%8C%DA%86%D8%B4" title="پیچش – Persian" lang="fa" hreflang="fa" data-title="پیچش" 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/Torsion" title="Torsion – French" lang="fr" hreflang="fr" data-title="Torsion" 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/Toirsi%C3%BAn" title="Toirsiún – Irish" lang="ga" hreflang="ga" data-title="Toirsiún" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B9%84%ED%8B%80%EB%A6%BC" title="비틀림 – Korean" lang="ko" hreflang="ko" data-title="비틀림" 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%88%D5%AC%D5%B8%D6%80%D5%B8%D6%82%D5%B4_(%D5%A4%D5%A5%D6%86%D5%B8%D6%80%D5%B4%D5%A1%D6%81%D5%AB%D5%A1)" title="Ոլորում (դեֆորմացիա) – Armenian" lang="hy" hreflang="hy" data-title="Ոլորում (դեֆորմացիա)" 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%AE%E0%A4%B0%E0%A5%8B%E0%A4%A1%E0%A4%BC" title="मरोड़ – Hindi" lang="hi" hreflang="hi" data-title="मरोड़" 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/Torzija" title="Torzija – Croatian" lang="hr" hreflang="hr" data-title="Torzija" 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/Puntiran" title="Puntiran – Indonesian" lang="id" hreflang="id" data-title="Puntiran" 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-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Torsion" title="Torsion – Interlingua" lang="ia" hreflang="ia" data-title="Torsion" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Torsione" title="Torsione – Italian" lang="it" hreflang="it" data-title="Torsione" 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%9E%D7%90%D7%9E%D7%A5_%D7%A4%D7%99%D7%AA%D7%95%D7%9C" title="מאמץ פיתול – Hebrew" lang="he" hreflang="he" data-title="מאמץ פיתול" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D0%BE%D1%80%D1%81%D0%B8%D0%BE%D0%BD" title="Торсион – Kazakh" lang="kk" hreflang="kk" data-title="Торсион" 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/V%C4%93rpes_deform%C4%81cija" title="Vērpes deformācija – Latvian" lang="lv" hreflang="lv" data-title="Vērpes deformācija" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wringspanning" title="Wringspanning – Dutch" lang="nl" hreflang="nl" data-title="Wringspanning" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Torsjon" title="Torsjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Torsjon" 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/Torsjon" title="Torsjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Torsjon" 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-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Buralish" title="Buralish – Uzbek" lang="uz" hreflang="uz" data-title="Buralish" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Skr%C4%99canie" title="Skręcanie – Polish" lang="pl" hreflang="pl" data-title="Skręcanie" 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/Tor%C3%A7%C3%A3o_mec%C3%A2nica" title="Torção mecânica – Portuguese" lang="pt" hreflang="pt" data-title="Torção mecânica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Torsiune_(mecanic%C4%83)" title="Torsiune (mecanică) – Romanian" lang="ro" hreflang="ro" data-title="Torsiune (mecanică)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D1%80%D1%83%D1%87%D0%B5%D0%BD%D0%B8%D0%B5_(%D0%B4%D0%B5%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D0%B8%D1%8F)" title="Кручение (деформация) – Russian" lang="ru" hreflang="ru" data-title="Кручение (деформация)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Torsion_(mechanics)" title="Torsion (mechanics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Torsion (mechanics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Vzvoj" title="Vzvoj – Slovenian" lang="sl" hreflang="sl" data-title="Vzvoj" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Uvijanje" title="Uvijanje – Serbian" lang="sr" hreflang="sr" data-title="Uvijanje" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Torzija" title="Torzija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Torzija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Vridning" title="Vridning – Swedish" lang="sv" hreflang="sv" data-title="Vridning" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B5%D1%84%D0%BE%D1%80%D0%BC%D0%B0%D1%86%D1%96%D1%8F_%D0%BA%D1%80%D1%83%D1%87%D0%B5%D0%BD%D0%BD%D1%8F" title="Деформація кручення – Ukrainian" lang="uk" hreflang="uk" data-title="Деформація кручення" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%89%AD%E8%BD%AC" title="扭转 – Chinese" lang="zh" hreflang="zh" data-title="扭转" data-language-autonym="中文" data-language-local-name="Chinese" 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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"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Twisting_force&redirect=no" class="mw-redirect" title="Twisting force">Twisting force</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><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">For other notions of torsion, see <a href="/wiki/Torsion_(disambiguation)" class="mw-redirect mw-disambig" title="Torsion (disambiguation)">Torsion (disambiguation)</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Twisting of an object due to an applied torque</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Twisted_bar.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Twisted_bar.png/220px-Twisted_bar.png" decoding="async" width="220" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Twisted_bar.png/330px-Twisted_bar.png 1.5x, //upload.wikimedia.org/wikipedia/commons/4/4f/Twisted_bar.png 2x" data-file-width="424" data-file-height="187" /></a><figcaption>Torsion of a square section bar</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Torsion_Blender.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Torsion_Blender.gif/220px-Torsion_Blender.gif" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/92/Torsion_Blender.gif/330px-Torsion_Blender.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/92/Torsion_Blender.gif/440px-Torsion_Blender.gif 2x" data-file-width="800" data-file-height="600" /></a><figcaption>Example of torsion mechanics</figcaption></figure> <p>In the field of <a href="/wiki/Solid_mechanics" title="Solid mechanics">solid mechanics</a>, <b>torsion</b> is the twisting of an object due to an applied <a href="/wiki/Torque" title="Torque">torque</a>. Torsion is expressed in either the <a href="/wiki/Pascal_(unit)" title="Pascal (unit)">pascal</a> (Pa), an <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a> unit for newtons per square metre, or in <a href="/wiki/Pounds_per_square_inch" class="mw-redirect" title="Pounds per square inch">pounds per square inch</a> (psi) while torque is expressed in <a href="/wiki/Newton_metre" class="mw-redirect" title="Newton metre">newton metres</a> (N·m) or <a href="/wiki/Foot-pound_force" class="mw-redirect" title="Foot-pound force">foot-pound force</a> (ft·lbf). In sections perpendicular to the torque axis, the resultant <a href="/wiki/Shear_stress" title="Shear stress">shear stress</a> in this section is perpendicular to the radius. </p><p>In non-circular cross-sections, twisting is accompanied by a distortion called warping, in which transverse sections do not remain plane.<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> For shafts of uniform cross-section unrestrained against warping, the torsion is: </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 T={\frac {J_{\text{T}}}{r}}\tau ={\frac {J_{\text{T}}}{\ell }}G\varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mi>r</mi> </mfrac> </mrow> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> <mi>ℓ</mi> </mfrac> </mrow> <mi>G</mi> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T={\frac {J_{\text{T}}}{r}}\tau ={\frac {J_{\text{T}}}{\ell }}G\varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f722cee06d54c595fbf3fd8793962d06a7c64a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.472ex; height:5.343ex;" alt="{\displaystyle T={\frac {J_{\text{T}}}{r}}\tau ={\frac {J_{\text{T}}}{\ell }}G\varphi }"></span></dd></dl> <p>where: </p> <ul><li><i>T</i> is the applied torque or moment of torsion in Nm.</li> <li><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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> (tau) is the maximum shear stress at the outer surface</li> <li><i>J</i><sub>T</sub> is the <a href="/wiki/Torsion_constant" title="Torsion constant">torsion constant</a> for the section. For circular rods, and tubes with constant wall thickness, it is equal to the polar moment of inertia of the section, but for other shapes, or split sections, it can be much less. For more accuracy, <a href="/wiki/Finite_element_method" title="Finite element method">finite element analysis</a> (FEA) is the best method. Other calculation methods include <a href="/wiki/Membrane_analogy" title="Membrane analogy">membrane analogy</a> and shear flow approximation.<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><i>r</i> is the perpendicular distance between the rotational axis and the farthest point in the section (at the outer surface).</li> <li><i>ℓ</i> is the length of the object to or over which the torque is being applied.</li> <li><i>φ</i> (phi) is the angle of twist in <a href="/wiki/Radian" title="Radian">radians</a>.</li> <li><i>G</i> is the shear modulus, also called the <a href="/wiki/Modulus_of_rigidity" class="mw-redirect" title="Modulus of rigidity">modulus of rigidity</a>, and is usually given in <a href="/wiki/Gigapascal" class="mw-redirect" title="Gigapascal">gigapascals</a> (GPa), <a href="/wiki/Pounds_per_square_inch" class="mw-redirect" title="Pounds per square inch">lbf/in<sup>2</sup></a> (psi), or lbf/ft<sup>2</sup> or in ISO units N/mm<sup>2</sup>.</li> <li>The product <i>J</i><sub>T</sub><i>G</i> is called the <a href="/wiki/Torsional_rigidity" class="mw-redirect" title="Torsional rigidity">torsional rigidity</a> <i>w</i><sub>T</sub>.</li></ul> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torsion_(mechanics)&action=edit&section=1" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The shear stress at a point within a shaft is: </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 \tau _{\varphi _{z}}(r)={Tr \over J_{\text{T}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mi>r</mi> </mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{\varphi _{z}}(r)={Tr \over J_{\text{T}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd4c3fffee82daa5282686ef5045fab214eb15de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.614ex; height:5.509ex;" alt="{\displaystyle \tau _{\varphi _{z}}(r)={Tr \over J_{\text{T}}}}"></span></dd></dl> <p>Note that the highest shear stress occurs on the surface of the shaft, where the radius is maximum. High stresses at the surface may be compounded by <a href="/wiki/Stress_concentrations" class="mw-redirect" title="Stress concentrations">stress concentrations</a> such as rough spots. Thus, shafts for use in high torsion are polished to a fine surface finish to reduce the maximum stress in the shaft and increase their service life. </p><p>The angle of twist can be found by using: </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 \varphi _{}={\frac {T\ell }{GJ_{\text{T}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mi>ℓ</mi> </mrow> <mrow> <mi>G</mi> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi _{}={\frac {T\ell }{GJ_{\text{T}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa9175ae75416a6e4468861f674219c381c0f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.637ex; height:5.843ex;" alt="{\displaystyle \varphi _{}={\frac {T\ell }{GJ_{\text{T}}}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Sample_calculation">Sample calculation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torsion_(mechanics)&action=edit&section=2" title="Edit section: Sample calculation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Dampfturbine_Laeufer01.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Dampfturbine_Laeufer01.jpg/250px-Dampfturbine_Laeufer01.jpg" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Dampfturbine_Laeufer01.jpg/375px-Dampfturbine_Laeufer01.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Dampfturbine_Laeufer01.jpg/500px-Dampfturbine_Laeufer01.jpg 2x" data-file-width="2126" data-file-height="2126" /></a><figcaption>The rotor of a modern <a href="/wiki/Steam_turbine" title="Steam turbine">steam turbine</a></figcaption></figure> <p>Calculation of the <a href="/wiki/Steam_turbine" title="Steam turbine">steam turbine</a> shaft radius for a turboset: </p><p>Assumptions: </p> <ul><li>Power carried by the shaft is 1000 <a href="/wiki/Megawatts" class="mw-redirect" title="Megawatts">MW</a>; this is typical for a large <a href="/wiki/Nuclear_power" title="Nuclear power">nuclear power</a> plant.</li> <li><a href="/wiki/Yield_stress" class="mw-redirect" title="Yield stress">Yield stress</a> of the steel used to make the shaft (<i>τ</i><sub>yield</sub>) is: 250 × 10<sup>6</sup> N/m<sup>2</sup>.</li> <li>Electricity has a frequency of 50 <a href="/wiki/Hertz" title="Hertz">Hz</a>; this is the typical frequency in Europe. In North America, the frequency is 60 Hz.</li></ul> <p>The <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> can be calculated with the following formula: </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 \omega =2\pi f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =2\pi f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1bf35d395c2d52391265e4bbda0aed14f52579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.317ex; height:2.509ex;" alt="{\displaystyle \omega =2\pi f}"></span></dd></dl> <p>The torque carried by the shaft is related to the <a href="/wiki/Power_(physics)" title="Power (physics)">power</a> by the following equation: </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 P=T\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>T</mi> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=T\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a740d3dd7c3b39b98d071bb000f7121f54dcdaa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.926ex; height:2.176ex;" alt="{\displaystyle P=T\omega }"></span></dd></dl> <p>The angular frequency is therefore 314.16 <a href="/wiki/Radian" title="Radian">rad</a>/<a href="/wiki/Second" title="Second">s</a> and the torque 3.1831 × 10<sup>6</sup> <a href="/wiki/Newton_metre" class="mw-redirect" title="Newton metre">N·m</a>. </p><p>The maximal torque is: </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 T_{\max }={\frac {{\tau }_{\max }J_{\text{zz}}}{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>zz</mtext> </mrow> </msub> </mrow> <mi>r</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{\max }={\frac {{\tau }_{\max }J_{\text{zz}}}{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e22525c3c32b98e5716df32305cf2a154895fd73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.873ex; height:5.176ex;" alt="{\displaystyle T_{\max }={\frac {{\tau }_{\max }J_{\text{zz}}}{r}}}"></span></dd></dl> <p>After substitution of the <i><a href="/wiki/Torsion_constant" title="Torsion constant">torsion constant</a></i>, the following expression is obtained: </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 D=\left({\frac {16T_{\max }}{\pi {\tau }_{\max }}}\right)^{1/3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>16</mn> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> </msub> </mrow> <mrow> <mi>π</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D=\left({\frac {16T_{\max }}{\pi {\tau }_{\max }}}\right)^{1/3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/236d336a6a5006165bfc598447335308042ef8f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.952ex; height:6.676ex;" alt="{\displaystyle D=\left({\frac {16T_{\max }}{\pi {\tau }_{\max }}}\right)^{1/3}}"></span></dd></dl> <p>The <a href="/wiki/Diameter" title="Diameter">diameter</a> is 40 cm. If one adds a <a href="/wiki/Factor_of_safety" title="Factor of safety">factor of safety</a> of 5 and re-calculates the radius with the maximum stress equal to the <i>yield stress/5</i>, the result is a diameter of 69 cm, the approximate size of a turboset shaft in a nuclear power plant. </p> <div class="mw-heading mw-heading2"><h2 id="Failure_mode">Failure mode</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Torsion_(mechanics)&action=edit&section=3" title="Edit section: Failure mode"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .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/Torsion_(mechanics)" title="Special:EditPage/Torsion (mechanics)">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=%22Torsion%22+mechanics">"Torsion" mechanics</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Torsion%22+mechanics+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Torsion%22+mechanics&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=%22Torsion%22+mechanics+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Torsion%22+mechanics">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Torsion%22+mechanics&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">December 2014</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>The shear stress in the shaft may be resolved into <a href="/wiki/Principal_stress" class="mw-redirect" title="Principal stress">principal stresses</a> via <a href="/wiki/Mohr%27s_circle" title="Mohr's circle">Mohr's circle</a>. If the shaft is loaded only in torsion, then one of the principal stresses will be in tension and the other in compression. These stresses are oriented at a 45-degree helical angle around the shaft. If the shaft is made of <a href="/wiki/Brittle" class="mw-redirect" title="Brittle">brittle</a> material, then the shaft will fail by a crack initiating at the surface and propagating through to the core of the shaft, fracturing in a 45-degree angle helical shape. This is often demonstrated by twisting a piece of blackboard chalk between one's fingers.<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> </p><p>In the case of thin hollow shafts, a twisting buckling mode can result from excessive torsional load, with wrinkles forming at 45° to the shaft axis. </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=Torsion_(mechanics)&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_area_moments_of_inertia" class="mw-redirect" title="List of area moments of inertia">List of area moments of inertia</a></li> <li><a href="/wiki/Saint-Venant%27s_theorem" title="Saint-Venant's theorem">Saint-Venant's theorem</a></li> <li><a href="/wiki/Second_moment_of_area" title="Second moment of area">Second moment of area</a></li> <li><a href="/wiki/Structural_rigidity" title="Structural rigidity">Structural rigidity</a></li> <li><a href="/wiki/Torque_tester" title="Torque tester">Torque tester</a></li> <li><a href="/wiki/Torsion_siege_engine" title="Torsion siege engine">Torsion siege engine</a></li> <li><a href="/wiki/Torsion_spring" title="Torsion spring">Torsion spring</a> or -bar</li> <li><a href="/wiki/Torsional_vibration" title="Torsional vibration">Torsional vibration</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=Torsion_(mechanics)&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><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="CITEREFSeaburgCarter1997" class="citation book cs1">Seaburg, Paul; Carter, Charles (1997). <i>Torsional Analysis of Structural Steel Members</i>. <a href="/wiki/American_Institute_of_Steel_Construction" title="American Institute of Steel Construction">American Institute of Steel Construction</a>. p. 3.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Torsional+Analysis+of+Structural+Steel+Members&rft.pages=3&rft.pub=American+Institute+of+Steel+Construction&rft.date=1997&rft.aulast=Seaburg&rft.aufirst=Paul&rft.au=Carter%2C+Charles&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorsion+%28mechanics%29" 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">Case and Chilver "Strength of Materials and Structures</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="CITEREFFakouri_HasanabadiKokabiFaghihi-SaniGroß-Barsnick2018" class="citation journal cs1">Fakouri Hasanabadi, M.; Kokabi, A.H.; Faghihi-Sani, M.A.; Groß-Barsnick, S.M.; Malzbender, J. (October 2018). "Room- and high-temperature torsional shear strength of solid oxide fuel/electrolysis cell sealing material". <i>Ceramics International</i>. <b>45</b> (2): 2219–2225. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.ceramint.2018.10.134">10.1016/j.ceramint.2018.10.134</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0272-8842">0272-8842</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:139371841">139371841</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ceramics+International&rft.atitle=Room-+and+high-temperature+torsional+shear+strength+of+solid+oxide+fuel%2Felectrolysis+cell+sealing+material&rft.volume=45&rft.issue=2&rft.pages=2219-2225&rft.date=2018-10&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A139371841%23id-name%3DS2CID&rft.issn=0272-8842&rft_id=info%3Adoi%2F10.1016%2Fj.ceramint.2018.10.134&rft.aulast=Fakouri+Hasanabadi&rft.aufirst=M.&rft.au=Kokabi%2C+A.H.&rft.au=Faghihi-Sani%2C+M.A.&rft.au=Gro%C3%9F-Barsnick%2C+S.M.&rft.au=Malzbender%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATorsion+%28mechanics%29" class="Z3988"></span></span> </li> </ol></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=Torsion_(mechanics)&action=edit&section=6" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/16px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/24px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/32px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> The dictionary definition of <a href="https://en.wiktionary.org/wiki/torsion" class="extiw" title="wiktionary:torsion"><i>torsion</i></a> at Wiktionary</li> <li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikibooks-logo-en-noslogan.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/16px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/24px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/32px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></a></span> <a href="https://en.wikibooks.org/wiki/Solid_Mechanics" class="extiw" title="wikibooks:Solid Mechanics">Solid Mechanics</a> at Wikibooks</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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