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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Trägheitsradius</span></h1>
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<p>Der <b>Trägheitsradius</b> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
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<mi>i</mi>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> ist derjenige Abstand von der gegebenen <a href="Drehachse" class="mw-redirect" title="Drehachse">Drehachse</a>, in dem man die <a href="Punktmasse" class="mw-redirect" title="Punktmasse">punktförmig gedachte Masse</a>&nbsp;m des Körpers anbringen muss, um das <a href="Tr%C3%A4gheitsmoment" title="Trägheitsmoment">Trägheitsmoment</a>&nbsp;J zu erhalten.<sup id="cite_ref-Boege_1-0" class="reference"><a href="#cite_note-Boege-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}J&amp;=m\cdot i^{2}\\\Leftrightarrow i^{2}&amp;={\frac {J}{m}}\\\Leftrightarrow i&amp;={\sqrt {\frac {J}{m}}}\end{aligned}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}J&amp;=m\cdot i^{2}\\\Leftrightarrow i^{2}&amp;={\frac {J}{m}}\\\Leftrightarrow i&amp;={\sqrt {\frac {J}{m}}}\end{aligned}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f52ad411c074b194e02c1857dbc63b8703b0193.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:14.252ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}J&amp;=m\cdot i^{2}\\\Leftrightarrow i^{2}&amp;={\frac {J}{m}}\\\Leftrightarrow i&amp;={\sqrt {\frac {J}{m}}}\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>Für eine gute Materialausnutzung z.&nbsp;B. bei <a href="Schwungrad" title="Schwungrad">Schwungrädern</a> wird ein Trägheitsradius angestrebt, der im Vergleich zur Außenabmessung groß ist, d.&nbsp;h. möglichst weit außen liegt.
</p><p>In der <a href="Festigkeitslehre" title="Festigkeitslehre">Festigkeitslehre</a> gilt ein analoger Zusammenhang zwischen der Fläche&nbsp;A und dem <a href="Fl%C3%A4chenmoment" title="Flächenmoment">Flächenmoment</a> zweiten Grades&nbsp;I:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}I&amp;=A\cdot i^{2}\\\Leftrightarrow i^{2}&amp;={\frac {I}{A}}\\\Leftrightarrow i&amp;={\sqrt {\frac {I}{A}}}\end{aligned}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}I&amp;=A\cdot i^{2}\\\Leftrightarrow i^{2}&amp;={\frac {I}{A}}\\\Leftrightarrow i&amp;={\sqrt {\frac {I}{A}}}\end{aligned}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4f8cd8a4bd4e829f1eecccc1e11a49473abfca5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:13.954ex; height:15.009ex;" alt="{\displaystyle {\begin{aligned}I&amp;=A\cdot i^{2}\\\Leftrightarrow i^{2}&amp;={\frac {I}{A}}\\\Leftrightarrow i&amp;={\sqrt {\frac {I}{A}}}\end{aligned}}}" loading="lazy"></span></dd></dl>
<p>Der Trägheitsradius geht hier als Berechnungsgröße in den Nachweis von <a href="Knicklast" class="mw-redirect" title="Knicklast">Knicklasten</a> ein.
</p>

<div class="mw-heading mw-heading2"><h2 id="Berechnung_des_Trägheitsradiuses_bei_bekannter_Geometrie"><span id="Berechnung_des_Tr.C3.A4gheitsradiuses_bei_bekannter_Geometrie"></span>Berechnung des Trägheitsradiuses bei bekannter Geometrie</h2></div>

<p>Zylinder: <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {r_{i}}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<msub>
<mi>r</mi>
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<mi>i</mi>
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<mo>=</mo>
<mn>0</mn>
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<annotation encoding="application/x-tex">{\displaystyle {r_{i}}=0}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8288ec7f8eb2dc6250db00924a25e011e9aa75ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.109ex; height:2.509ex;" alt="{\displaystyle {r_{i}}=0}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=?}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>=</mo>
<mo>?</mo>
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<annotation encoding="application/x-tex">{\displaystyle i=?}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/220729b142cf36be9621511561207e498d7cc782.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.353ex; height:2.176ex;" alt="{\displaystyle i=?}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {r_{a}}\,{\text{bekannt}}}">
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<mspace width="thinmathspace"></mspace>
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<mtext>bekannt</mtext>
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<annotation encoding="application/x-tex">{\displaystyle {r_{a}}\,{\text{bekannt}}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c68f45f57a90f1849b6e2c0d56d7464faf494dea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.742ex; height:2.509ex;" alt="{\displaystyle {r_{a}}\,{\text{bekannt}}}" loading="lazy"></span><br>
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}A_{1}&amp;=A_{2}\\\Leftrightarrow {i^{2}\cdot \pi }&amp;={r_{a}}^{2}\cdot \pi -i^{2}\cdot \pi \\\Leftrightarrow i&amp;={\sqrt {\frac {{r_{a}}^{2}}{2}}}\\\end{aligned}}}">
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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}A_{1}&amp;=A_{2}\\\Leftrightarrow {i^{2}\cdot \pi }&amp;={r_{a}}^{2}\cdot \pi -i^{2}\cdot \pi \\\Leftrightarrow i&amp;={\sqrt {\frac {{r_{a}}^{2}}{2}}}\\\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7bf29e6f08756f1d0203e3803ffbbca1dbe2d1ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.396ex; margin-bottom: -0.275ex; width:25.611ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}A_{1}&amp;=A_{2}\\\Leftrightarrow {i^{2}\cdot \pi }&amp;={r_{a}}^{2}\cdot \pi -i^{2}\cdot \pi \\\Leftrightarrow i&amp;={\sqrt {\frac {{r_{a}}^{2}}{2}}}\\\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>beliebiger Querschnitt:<br>
<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{1}=A_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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</msub>
<mo>=</mo>
<msub>
<mi>A</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msub>
</mstyle>
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<annotation encoding="application/x-tex">{\displaystyle A_{1}=A_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ccb4687bbe819cd8e1f6c6afb06b9316a6c18b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.693ex; height:2.509ex;" alt="{\displaystyle A_{1}=A_{2}}" loading="lazy"></span> → Aus dieser Gleichung lässt sich jeweils der Trägheitsradius errechnen.<br>
</p>
<div class="mw-heading mw-heading2"><h2 id="Polymerchemie">Polymerchemie</h2></div>
<p>In der <a href="Polymerchemie" title="Polymerchemie">Polymerchemie</a> wird der mittlere quadratische Abstand der Molekülketten vom <a href="Massenmittelpunkt" title="Massenmittelpunkt">Schwerpunkt</a> des <a href="Molek%C3%BCl" title="Molekül">Moleküls</a> als <a href="Streumassenradius" title="Streumassenradius">Streumassenradius</a> bezeichnet, teilweise auch als Trägheitsradius.<sup id="cite_ref-Schwarzl_2-0" class="reference"><a href="#cite_note-Schwarzl-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Martin Mayr: <cite style="font-style:italic">Technische Mechanik: Statik – Kinematik – Kinetik – Schwingungen – Festigkeitslehre</cite>. Teil 3, ISBN 978-3-446-22608-1, <span style="white-space:nowrap">Kap.<span style="display:inline-block;width:.2em">&nbsp;</span>5</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abookitem&amp;rfr_id=info:sid/de.wikipedia.org:Tr%C3%A4gheitsradius&amp;rft.atitle=5&amp;rft.au=Martin+Mayr&amp;rft.btitle=Technische+Mechanik%3A+Statik+-+Kinematik+-+Kinetik+-+Schwingungen+-+Festigkeitslehre&amp;rft.genre=bookitem&amp;rft.isbn=9783446226081&amp;rft.volume=Teil+3" style="display:none">&nbsp;</span></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-Boege-1"><span class="mw-cite-backlink"><a href="#cite_ref-Boege_1-0">↑</a></span> <span class="reference-text">
Alfred Böge (Hrsg.): <cite style="font-style:italic">Vieweg Handbuch Maschinenbau: Grundlagen und Anwendungen der Maschinenbau-Technik</cite>. 18. Auflage. Vieweg, 2007, ISBN 978-3-8348-0110-4 (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=vsSpxaCNS_QC&amp;pg=RA1-PA68#v=onepage">eingeschränkte Vorschau</a> in der Google-Buchsuche).<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:Tr%C3%A4gheitsradius&amp;rft.btitle=Vieweg+Handbuch+Maschinenbau%3A+Grundlagen+und+Anwendungen+der+Maschinenbau-Technik&amp;rft.date=2007&amp;rft.edition=18.&amp;rft.genre=book&amp;rft.isbn=9783834801104&amp;rft.pub=Vieweg" style="display:none">&nbsp;</span></span>
</li>
<li id="cite_note-Schwarzl-2"><span class="mw-cite-backlink"><a href="#cite_ref-Schwarzl_2-0">↑</a></span> <span class="reference-text">
Friedrich R. Schwarzl: <cite style="font-style:italic">Polymermechanik: Struktur und mechanisches Verhalten von Polymeren</cite>. Springer, 1990, ISBN 3-540-51965-3 (<a rel="nofollow" class="external text" href="https://books.google.de/books?id=MLwft8SayvQC&amp;pg=PA71#v=onepage">eingeschränkte Vorschau</a> in der Google-Buchsuche).<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:Tr%C3%A4gheitsradius&amp;rft.au=Friedrich+R.+Schwarzl&amp;rft.btitle=Polymermechanik%3A+Struktur+und+mechanisches+Verhalten+von+Polymeren&amp;rft.date=1990&amp;rft.genre=book&amp;rft.isbn=3540519653&amp;rft.pub=Springer" style="display:none">&nbsp;</span></span>
</li>
</ol>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li><a rel="nofollow" class="external text" href="http://www.moroder.it/daniel/data/GEO%20Festigkeitslehre.pdf">Festigkeitslehre</a> (abgerufen am 18. Oktober 2019)</li>
<li><a rel="nofollow" class="external text" href="http://wandinger.userweb.mwn.de/TM2/v3_2.pdf">Flächenträgheitsmomente</a> (abgerufen am 18. Oktober 2019)</li>
<li><a rel="nofollow" class="external text" href="https://home.uni-leipzig.de/prakphys/pdf/VersuchePhy/Mechanik/M-13-AUF.pdf">Fakultät für Physik und Geowissenschaften Physikalisches Grundpraktikum</a> (abgerufen am 18. Oktober 2019)</li>
<li><a rel="nofollow" class="external text" href="http://www-brs.ub.ruhr-uni-bochum.de/netahtml/HSS/Diss/WolfChristian/diss.pdf">Tragfähigkeit von Stäben aus Baustahl – Nichtlineares Tragverhalten, Stabilität, Nachweisverfahren</a> (abgerufen am 18. Oktober 2019)</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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