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<span>References</span> </div> </a> <ul id="toc-References-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 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href="https://cs.wikipedia.org/wiki/Neutr%C3%A1ln%C3%AD_osa" title="Neutrální osa – Czech" lang="cs" hreflang="cs" data-title="Neutrální osa" 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/Neutrale_Faser" title="Neutrale Faser – German" lang="de" hreflang="de" data-title="Neutrale Faser" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Fibra_neutra" title="Fibra neutra – Spanish" lang="es" hreflang="es" data-title="Fibra neutra" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><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/Neutral_axis" title="Special:EditPage/Neutral axis">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=%22Neutral+axis%22">"Neutral axis"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Neutral+axis%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Neutral+axis%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Neutral+axis%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Neutral+axis%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Neutral+axis%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">February 2023</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> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Bending.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Bending.svg/220px-Bending.svg.png" decoding="async" width="220" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Bending.svg/330px-Bending.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Bending.svg/440px-Bending.svg.png 2x" data-file-width="320" data-file-height="190" /></a><figcaption>Beam with neutral axis (x).</figcaption></figure> <p>The <b>neutral axis</b> is an axis in the cross section of a <a href="/wiki/Beam_(structure)" title="Beam (structure)">beam</a> (a member resisting bending) or shaft along which there are no longitudinal stresses or strains. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Theory">Theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Neutral_axis&action=edit&section=1" title="Edit section: Theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the section is symmetric, isotropic and is not curved before a bend occurs, then the neutral axis is at the geometric <a href="/wiki/Centroid" title="Centroid">centroid</a> of a beam or shaft. All fibers on one side of the neutral axis are in a state of <a href="/wiki/Tension_(physics)" title="Tension (physics)">tension</a>, while those on the opposite side are in <a href="/wiki/Compression_(physical)" class="mw-redirect" title="Compression (physical)">compression</a>. </p><p>Since the beam is undergoing uniform bending, a plane on the beam remains plane. That is: </p><p><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 \gamma _{xy}=\gamma _{zx}=\tau _{xy}=\tau _{xz}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{xy}=\gamma _{zx}=\tau _{xy}=\tau _{xz}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1de2be68f27c8a5f9059511618e2aa76f428c289" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.86ex; height:2.843ex;" alt="{\displaystyle \gamma _{xy}=\gamma _{zx}=\tau _{xy}=\tau _{xz}=0}"></span> </p><p>Where <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 \gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }"></span> is the <a href="/wiki/Shear_strain" class="mw-redirect" title="Shear strain">shear strain</a> and <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> is the <a href="/wiki/Shear_stress" title="Shear stress">shear stress</a> </p><p>There is a compressive (negative) strain at the top of the beam, and a tensile (positive) strain at the bottom of the beam. Therefore by the <a href="/wiki/Continuous_function#Intermediate_value_theorem" title="Continuous function">Intermediate Value Theorem</a>, there must be some point in between the top and the bottom that has no strain, since the strain in a beam is a <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a>. </p><p>Let L be the original length of the beam (<a href="/wiki/Span_(architecture)" class="mw-redirect" title="Span (architecture)">span</a>)<br /> ε(y) is the strain as a function of coordinate on the face of the beam.<br /> σ(y) is the stress as a function of coordinate on the face of the beam.<br /> ρ is the <a href="/wiki/Radius_of_curvature_(applications)" class="mw-redirect" title="Radius of curvature (applications)">radius of curvature</a> of the beam at its neutral axis.<br /> θ is the <a href="/wiki/Bending" title="Bending">bend</a> <a href="/wiki/Angle" title="Angle">angle</a> </p><p>Since the bending is <a href="/wiki/Uniform" title="Uniform">uniform</a> and pure, there is therefore at a distance y from the neutral axis with the inherent property of having no strain: </p><p><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 \epsilon _{x}(y)={\frac {L(y)-L}{L}}={\frac {\theta \,(\rho \,-y)-\theta \rho \,}{\theta \rho \,}}={\frac {-y\theta }{\rho \theta }}={\frac {-y}{\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>L</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>L</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>θ</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>ρ</mi> <mspace width="thinmathspace" /> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>θ</mi> <mi>ρ</mi> <mspace width="thinmathspace" /> </mrow> <mrow> <mi>θ</mi> <mi>ρ</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>y</mi> <mi>θ</mi> </mrow> <mrow> <mi>ρ</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>y</mi> </mrow> <mi>ρ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{x}(y)={\frac {L(y)-L}{L}}={\frac {\theta \,(\rho \,-y)-\theta \rho \,}{\theta \rho \,}}={\frac {-y\theta }{\rho \theta }}={\frac {-y}{\rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc3c43ffd199624b2915e3e0b8496c5cfa0a1c33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.2ex; height:6.343ex;" alt="{\displaystyle \epsilon _{x}(y)={\frac {L(y)-L}{L}}={\frac {\theta \,(\rho \,-y)-\theta \rho \,}{\theta \rho \,}}={\frac {-y\theta }{\rho \theta }}={\frac {-y}{\rho }}}"></span> </p><p>Therefore the longitudinal normal strain <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 \epsilon _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41af29de2b1d4b9ac93f30b80ff3d21e3a10e32f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.117ex; height:2.009ex;" alt="{\displaystyle \epsilon _{x}}"></span> varies linearly with the distance y from the neutral surface. Denoting <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 \epsilon _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acd3a15e19c6c4fcae9137cc850a5e1705c6576f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.619ex; height:2.009ex;" alt="{\displaystyle \epsilon _{m}}"></span> as the maximum strain in the beam (at a distance c from the neutral axis), it becomes clear that: </p><p><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 \epsilon _{m}={\frac {c}{\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>ρ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{m}={\frac {c}{\rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae34f148c8bb6d5ea506b78643e8ad8b4464ae10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.756ex; height:5.176ex;" alt="{\displaystyle \epsilon _{m}={\frac {c}{\rho }}}"></span> </p><p>Therefore, we can solve for ρ, and find that: </p><p><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 \rho ={\frac {c}{\epsilon _{m}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {c}{\epsilon _{m}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/880b9958bb55502b7f63fc68454ec38c0999b4d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:7.756ex; height:5.009ex;" alt="{\displaystyle \rho ={\frac {c}{\epsilon _{m}}}}"></span> </p><p>Substituting this back into the original expression, we find that: </p><p><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 \epsilon _{x}(y)={\frac {-\epsilon _{m}y}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>y</mi> </mrow> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{x}(y)={\frac {-\epsilon _{m}y}{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4834334fb6be64a067c51ce05a0f732e301dd3cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.599ex; height:5.176ex;" alt="{\displaystyle \epsilon _{x}(y)={\frac {-\epsilon _{m}y}{c}}}"></span> </p><p>Due to <a href="/wiki/Hooke%27s_Law" class="mw-redirect" title="Hooke's Law">Hooke's Law</a>, the stress in the beam is proportional to the strain by E, the <a href="/wiki/Modulus_of_elasticity" class="mw-redirect" title="Modulus of elasticity">modulus of elasticity</a>: </p><p><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 _{x}=E\epsilon _{x}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mi>E</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}=E\epsilon _{x}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcb995dd8fafabfa75cc891c3a97e778bbb7a249" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.878ex; height:2.509ex;" alt="{\displaystyle \sigma _{x}=E\epsilon _{x}\,}"></span> </p><p>Therefore: </p><p><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 E\epsilon _{x}(y)={\frac {-E\epsilon _{m}y}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>E</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>y</mi> </mrow> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\epsilon _{x}(y)={\frac {-E\epsilon _{m}y}{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f940571b37cabfa6ed14f06d52dc7f4fa921329" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.15ex; height:5.343ex;" alt="{\displaystyle E\epsilon _{x}(y)={\frac {-E\epsilon _{m}y}{c}}}"></span> </p><p><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 _{x}(y)={\frac {-\sigma _{m}y}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>y</mi> </mrow> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{x}(y)={\frac {-\sigma _{m}y}{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67122a98af689858be4456856386ed8e9bd67df3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.365ex; height:5.176ex;" alt="{\displaystyle \sigma _{x}(y)={\frac {-\sigma _{m}y}{c}}}"></span> </p><p>From <a href="/wiki/Statics" title="Statics">statics</a>, a moment (i.e. <a href="/wiki/Pure_bending" title="Pure bending">pure bending</a>) consists of equal and opposite forces. Therefore, the total amount of force across the cross section must be 0. </p><p><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 \int \sigma _{x}dA=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \sigma _{x}dA=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/889dde3418326f241322c56d729a10245e637e7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.301ex; height:5.676ex;" alt="{\displaystyle \int \sigma _{x}dA=0}"></span> </p><p>Therefore: </p><p><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 \int {\frac {-\sigma _{m}y}{c}}dA=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>y</mi> </mrow> <mi>c</mi> </mfrac> </mrow> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\frac {-\sigma _{m}y}{c}}dA=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3477213e7879a79bafa24a5a521cc52e517c4dc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.603ex; height:5.676ex;" alt="{\displaystyle \int {\frac {-\sigma _{m}y}{c}}dA=0}"></span> </p><p>Since y denotes the distance from the neutral axis to any point on the face, it is the only variable that changes with respect to dA. Therefore: </p><p><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 \int ydA=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>y</mi> <mi>d</mi> <mi>A</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int ydA=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f58625f15eac013c78eddb04d2127a92db3ff83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.956ex; height:5.676ex;" alt="{\displaystyle \int ydA=0}"></span> </p><p>Therefore the <a href="/wiki/First_moment" class="mw-redirect" title="First moment">first moment</a> of the cross section about its neutral axis must be zero. Therefore the neutral axis lies on the centroid of the cross section. </p><p>Note that the neutral axis does not change in length when under bending. It may seem counterintuitive at first, but this is because there are no bending stresses in the neutral axis. However, there are shear stresses (τ) in the neutral axis, zero in the middle of the span but increasing towards the supports, as can be seen in this function (Jourawski's 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 \tau ={\frac {TQ}{wI}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>T</mi> <mi>Q</mi> </mrow> <mrow> <mi>w</mi> <mi>I</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau ={\frac {TQ}{wI}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b39309f98400d14be8fe8fa453716fb11f3f2d50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.611ex; height:5.343ex;" alt="{\displaystyle \tau ={\frac {TQ}{wI}}}"></span></dd></dl> <p>where<br /> T = shear force<br /> Q = <a href="/wiki/First_moment_of_area" title="First moment of area">first moment of area</a> of the section above/below the neutral axis<br /> w = width of the beam<br /> I = <a href="/wiki/Second_moment_of_area" title="Second moment of area">second moment of area</a> of the beam </p><p>This definition is suitable for the so-called long beams, i.e. its length is much larger than the other two dimensions. </p> <div class="mw-heading mw-heading2"><h2 id="Arches">Arches</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Neutral_axis&action=edit&section=2" title="Edit section: Arches"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Arch" title="Arch">Arches</a> also have a neutral axis if they are made of stone; stone is an inelastic medium, and has little strength in tension. Therefore as the loading on the arch changes the neutral axis moves- if the neutral axis leaves the stonework, then the arch will fail. </p><p>This theory (also known as the <a href="/wiki/Line_of_thrust" title="Line of thrust">thrust line</a> method) was proposed by <a href="/wiki/Thomas_Young_(scientist)" title="Thomas Young (scientist)">Thomas Young</a> and developed by <a href="/wiki/Isambard_Kingdom_Brunel" title="Isambard Kingdom Brunel">Isambard Kingdom Brunel</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Practical_applications">Practical applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Neutral_axis&action=edit&section=3" title="Edit section: Practical applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Building trades workers should have at least a basic understanding of the concept of neutral axis, to avoid cutting openings to route wires, pipes, or ducts in locations which may dangerously compromise the strength of structural elements of a building. <a href="/wiki/Building_code" title="Building code">Building codes</a> usually specify rules and guidelines which may be followed for routine work, but special situations and designs may need the services of a <a href="/wiki/Structural_engineer" title="Structural engineer">structural engineer</a> to assure safety.<sup id="cite_ref-ICC_1-0" class="reference"><a href="#cite_note-ICC-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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> </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=Neutral_axis&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/Neutral_plane" title="Neutral plane">Neutral plane</a></li> <li><a href="/wiki/Second_moment_of_inertia" class="mw-redirect" title="Second moment of inertia">Second moment of inertia</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=Neutral_axis&action=edit&section=5" 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"><ol class="references"> <li id="cite_note-ICC-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-ICC_1-0">^</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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://codes.iccsafe.org/s/IPC2015_NY/appendix-c-structural-safety/IPC2015-AppxC">"Digital Codes"</a>. <i>ICC Codes</i>. International Code Council, Inc<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-02-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ICC+Codes&rft.atitle=Digital+Codes&rft_id=https%3A%2F%2Fcodes.iccsafe.org%2Fs%2FIPC2015_NY%2Fappendix-c-structural-safety%2FIPC2015-AppxC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANeutral+axis" 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="CITEREFYehHerzog" class="citation web cs1">Yeh, Borjen; Herzog, Benjamin. <a rel="nofollow" class="external text" href="https://www.apawood.org/Data/Sites/1/documents/technicalresearch/MAT-012-04_Effect-of-Holes-on-the-Structural-Capacities-of-LVL_FullPaper.pdf">"Effect of holes on the structural capacities of laminated veneer lumber"</a> <span class="cs1-format">(PDF)</span>. <i>APA Wood</i>. APA – The Engineered Wood Association<span class="reference-accessdate">. Retrieved <span class="nowrap">2023-02-10</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=APA+Wood&rft.atitle=Effect+of+holes+on+the+structural+capacities+of+laminated+veneer+lumber&rft.aulast=Yeh&rft.aufirst=Borjen&rft.au=Herzog%2C+Benjamin&rft_id=https%3A%2F%2Fwww.apawood.org%2FData%2FSites%2F1%2Fdocuments%2Ftechnicalresearch%2FMAT-012-04_Effect-of-Holes-on-the-Structural-Capacities-of-LVL_FullPaper.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANeutral+axis" class="Z3988"></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Neutral_axis&oldid=1169723865">https://en.wikipedia.org/w/index.php?title=Neutral_axis&oldid=1169723865</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Boilermaking" title="Category:Boilermaking">Boilermaking</a></li><li><a href="/wiki/Category:Solid_mechanics" title="Category:Solid mechanics">Solid mechanics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_needing_additional_references_from_February_2023" title="Category:Articles needing additional references from February 2023">Articles needing additional references from February 2023</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 10 August 2023, at 21:52<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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