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<p>\begin{align}<br />I_{y}=L^{3} t(1-\cos (2 \alpha)) / 24 <br />\end{align}</p>
<p>\begin{align}<br />I_{z}=L^{3} t(1+\cos (2 \alpha)) / 24 <br />\end{align}</p>
<p>\begin{align}<br />I_{y z}=-L^{3} t \sin (2 \alpha) / 24<br />\end{align}</p>

<p>Im $\eta\zeta$-Koordinatensystem gilt:</p>


<p>Im <mjx-container class="MathJax" jax="SVG"><svg alt="\eta\zeta" style="vertical-align: -0.489ex" xmlns="http://www.w3.org/2000/svg" width="2.19ex" height="2.081ex" role="img" focusable="false" viewBox="0 -704 968 920" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-1070-TEX-I-1D702" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q156 442 175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336V326Q503 302 439 53Q381 -182 377 -189Q364 -216 332 -216Q319 -216 310 -208T299 -186Q299 -177 358 57L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-1070-TEX-I-1D701" d="M296 643Q298 704 324 704Q342 704 342 687Q342 682 339 664T336 633Q336 623 337 618T338 611Q339 612 341 612Q343 614 354 616T374 618L384 619H394Q471 619 471 586Q467 548 386 546H372Q338 546 320 564L311 558Q235 506 175 398T114 190Q114 171 116 155T125 127T137 104T153 86T171 72T192 61T213 53T235 46T256 39L322 16Q389 -10 389 -80Q389 -119 364 -154T300 -202Q292 -204 274 -204Q247 -204 225 -196Q210 -192 193 -182T172 -167Q167 -159 173 -148Q180 -139 191 -139Q195 -139 221 -153T283 -168Q298 -166 310 -152T322 -117Q322 -91 302 -75T250 -51T183 -29T116 4T65 62T44 160Q44 287 121 410T293 590L302 595Q296 613 296 643Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D702" xlink:href="#MJX-1070-TEX-I-1D702"></use></g><g data-mml-node="mi" transform="translate(497,0)"><use data-c="1D701" xlink:href="#MJX-1070-TEX-I-1D701"></use></g></g></g></svg></mjx-container>-Koordinatensystem gilt:</p>


<p>Das $ y z $ -Koordinatensystem geht durch Drehung um den Winkel $ -\alpha $ aus dem $ \eta \zeta $Koordinatensystem hervor.</p>


<p>Das <mjx-container class="MathJax" jax="SVG"><svg alt=" y z " style="vertical-align: -0.464ex" xmlns="http://www.w3.org/2000/svg" width="2.161ex" height="1.464ex" role="img" focusable="false" viewBox="0 -442 955 647" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-1072-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-1072-TEX-I-1D467" d="M347 338Q337 338 294 349T231 360Q211 360 197 356T174 346T162 335T155 324L153 320Q150 317 138 317Q117 317 117 325Q117 330 120 339Q133 378 163 406T229 440Q241 442 246 442Q271 442 291 425T329 392T367 375Q389 375 411 408T434 441Q435 442 449 442H462Q468 436 468 434Q468 430 463 420T449 399T432 377T418 358L411 349Q368 298 275 214T160 106L148 94L163 93Q185 93 227 82T290 71Q328 71 360 90T402 140Q406 149 409 151T424 153Q443 153 443 143Q443 138 442 134Q425 72 376 31T278 -11Q252 -11 232 6T193 40T155 57Q111 57 76 -3Q70 -11 59 -11H54H41Q35 -5 35 -2Q35 13 93 84Q132 129 225 214T340 322Q352 338 347 338Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-1072-TEX-I-1D466"></use></g><g data-mml-node="mi" transform="translate(490,0)"><use data-c="1D467" xlink:href="#MJX-1072-TEX-I-1D467"></use></g></g></g></svg></mjx-container> -Koordinatensystem geht durch Drehung um den Winkel <mjx-container class="MathJax" jax="SVG"><svg alt=" -\alpha " style="vertical-align: -0.186ex" xmlns="http://www.w3.org/2000/svg" width="3.208ex" height="1.505ex" role="img" focusable="false" viewBox="0 -583 1418 665" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-1073-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-1073-TEX-I-1D6FC" d="M34 156Q34 270 120 356T309 442Q379 442 421 402T478 304Q484 275 485 237V208Q534 282 560 374Q564 388 566 390T582 393Q603 393 603 385Q603 376 594 346T558 261T497 161L486 147L487 123Q489 67 495 47T514 26Q528 28 540 37T557 60Q559 67 562 68T577 70Q597 70 597 62Q597 56 591 43Q579 19 556 5T512 -10H505Q438 -10 414 62L411 69L400 61Q390 53 370 41T325 18T267 -2T203 -11Q124 -11 79 39T34 156ZM208 26Q257 26 306 47T379 90L403 112Q401 255 396 290Q382 405 304 405Q235 405 183 332Q156 292 139 224T121 120Q121 71 146 49T208 26Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-1073-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(778,0)"><use data-c="1D6FC" xlink:href="#MJX-1073-TEX-I-1D6FC"></use></g></g></g></svg></mjx-container> aus dem <mjx-container class="MathJax" jax="SVG"><svg alt=" \eta \zeta " style="vertical-align: -0.489ex" xmlns="http://www.w3.org/2000/svg" width="2.19ex" height="2.081ex" role="img" focusable="false" viewBox="0 -704 968 920" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-1074-TEX-I-1D702" d="M21 287Q22 290 23 295T28 317T38 348T53 381T73 411T99 433T132 442Q156 442 175 435T205 417T221 395T229 376L231 369Q231 367 232 367L243 378Q304 442 382 442Q436 442 469 415T503 336V326Q503 302 439 53Q381 -182 377 -189Q364 -216 332 -216Q319 -216 310 -208T299 -186Q299 -177 358 57L420 307Q423 322 423 345Q423 404 379 404H374Q288 404 229 303L222 291L189 157Q156 26 151 16Q138 -11 108 -11Q95 -11 87 -5T76 7T74 17Q74 30 114 189T154 366Q154 405 128 405Q107 405 92 377T68 316T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-1074-TEX-I-1D701" d="M296 643Q298 704 324 704Q342 704 342 687Q342 682 339 664T336 633Q336 623 337 618T338 611Q339 612 341 612Q343 614 354 616T374 618L384 619H394Q471 619 471 586Q467 548 386 546H372Q338 546 320 564L311 558Q235 506 175 398T114 190Q114 171 116 155T125 127T137 104T153 86T171 72T192 61T213 53T235 46T256 39L322 16Q389 -10 389 -80Q389 -119 364 -154T300 -202Q292 -204 274 -204Q247 -204 225 -196Q210 -192 193 -182T172 -167Q167 -159 173 -148Q180 -139 191 -139Q195 -139 221 -153T283 -168Q298 -166 310 -152T322 -117Q322 -91 302 -75T250 -51T183 -29T116 4T65 62T44 160Q44 287 121 410T293 590L302 595Q296 613 296 643Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D702" xlink:href="#MJX-1074-TEX-I-1D702"></use></g><g data-mml-node="mi" transform="translate(497,0)"><use data-c="1D701" xlink:href="#MJX-1074-TEX-I-1D701"></use></g></g></g></svg></mjx-container>Koordinatensystem hervor.</p>


<p>Die Transformationsformeln lauten:</p>


<p>\begin{align}<br />I_{y}&amp;=\frac{I_{\eta}+I_{\zeta}}{2}+\frac{I_{\eta}-I_{\eta}}{2} \cos (-2 \alpha)+I_{\eta \eta} \sin (-2 \alpha)=\frac{L^{3} t}{24}(1-\cos (2 \alpha)) \\\\<br />I_{z}&amp;=\frac{I_{\eta}+I_{\xi}}{2}-\frac{I_{\eta}-I_{\xi}}{2} \cos (-2 \alpha)-I_{\eta \xi} \sin (-2 \alpha)=\frac{L^{3} t}{24}(1+\cos (2 \alpha)) \\\\<br />I_{y z}&amp;=-\frac{I_{\eta}-I_{\zeta}}{2} \sin (-2 \alpha)+I_{\eta \eta} \cos (-2 \alpha)=-\frac{L^{3} t}{24} \sin (2 \alpha) \end{align}</p>


<p>\begin{array}{l} y=s \cos (\alpha) \\ z=s \sin (\alpha) \\ d A=t d s \end{array}</p>


<p>\begin{align}<br />I_{y}&amp;=\int_{-L / 2}^{L / 2} z^{2} \ t \ d s=\int_{-L / 2}^{L / 2} s^{2} \ ds \ t \ \sin ^{2}(\alpha) \\\\<br />&amp;=t\ \sin ^{2}(\alpha)\left[\frac{s^{3}}{3}\right]_{s=-L / 2}^{s=L / 2}\\\\<br />&amp;=\frac{L^{3} t}{12} \sin ^{2}(\alpha)<br />\end{align}</p>


<p>\begin{align}<br />I_{z}&amp;=\int_{-L / 2}^{L / 2} y^{2} \ t \ d s=\int_{-L / 2}^{L / 2} s^{2} ds \ t \cos ^{2}(\alpha) \\\\<br />&amp;=\frac{L^{3} t}{12} \cos ^{2}(\alpha)<br />\end{align}</p>


<p>\begin{align}<br />I_{y z}&amp;=-\int_{-L / 2}^{L / 2} y \  z \ t \ d s=-\int_{-L / 2}^{L / 2} s^{2} \ d s \ t \ \sin (\alpha) \cos (\alpha) \\\\<br />&amp;=-\frac{L^{3} t}{12} \sin (\alpha) \cos (\alpha)<br />\end{align}</p>


<p>Mit $ \sin ^{2}(\alpha)=\frac{1}{2}(1-\cos (2 \alpha)), \ \cos ^{2}(\alpha)=\frac{1}{2}(1+\cos (2 \alpha))$ und $\sin (\alpha) \cos (\alpha)=\frac{1}{2} \sin (2 \alpha)$ folgt:</p>


<p>Mit <mjx-container class="MathJax" jax="SVG"><svg alt=" \sin ^{2}(\alpha)=\frac{1}{2}(1-\cos (2 \alpha)), \ \cos ^{2}(\alpha)=\frac{1}{2}(1+\cos (2 \alpha))" style="vertical-align: -0.781ex" xmlns="http://www.w3.org/2000/svg" width="51.818ex" height="2.742ex" role="img" focusable="false" viewBox="0 -867 22903.5 1212" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-1080-TEX-N-73" d="M295 316Q295 356 268 385T190 414Q154 414 128 401Q98 382 98 349Q97 344 98 336T114 312T157 287Q175 282 201 278T245 269T277 256Q294 248 310 236T342 195T359 133Q359 71 321 31T198 -10H190Q138 -10 94 26L86 19L77 10Q71 4 65 -1L54 -11H46H42Q39 -11 33 -5V74V132Q33 153 35 157T45 162H54Q66 162 70 158T75 146T82 119T101 77Q136 26 198 26Q295 26 295 104Q295 133 277 151Q257 175 194 187T111 210Q75 227 54 256T33 318Q33 357 50 384T93 424T143 442T187 447H198Q238 447 268 432L283 424L292 431Q302 440 314 448H322H326Q329 448 335 442V310L329 304H301Q295 310 295 316Z"></path><path id="MJX-1080-TEX-N-69" d="M69 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<p>\begin{align}<br />I_{y}&amp;=\frac{L^{3} t}{24}(1-\cos (2 \alpha)) \\\\<br />I_{z}&amp;=\frac{L^{3} t}{24}(1+\cos (2 \alpha)) \\\\<br />I_{y z}&amp;=-\frac{L^{3} t}{24} \sin (2 \alpha)<br />\end{align}</p>

<div class="horizontalLayoutContainer" style="display: flex; flex-direction: row;">
<div class="horizontalLayoutElement" style="padding-right: 10px; flex: 1;">
<p>Für das abgebildete dünnwandige Profil sind zu berechnen:</p>
<ol style="list-style-type: lower-alpha;">
<li>die Flächenträgheitsmomente $ I_{y}, I_{z} $ und $ I_{y z} $ bezüglich dem eingezeichneten Koordinatensystem</li>
<li>die Hauptträgheitsmomente $ I_{1} $ und $ I_{2} $ sowie die Richtung $ \phi_{1} $ der ersten Hauptachse</li>
</ol>
<p>Die Ergebnisse sind mit dem Mohrschen Trägheitskreis zu überprüfen $ \left(a^{3} t / 6 \triangleq 1 \mathrm{~cm}\right) $.</p>
<p> </p>
<p><strong>Gegeben:</strong> $ a, t, e_{y}=a / 4 $</p>
</div>
<div class="horizontalLayoutElement" style="padding-left: 10px; flex: 1;">
<p><img style="width: 100%;" src="https://storage.googleapis.com/max-academy-image-bucket/602f9f0b2fe7b3a4e586f79e.png" alt="Abbildung" /></p>
</div>
</div>
<p> </p>

Dies ist eine interaktive Aufgabe zu: Flächenträgheitsmomente mit praktischen Tipps zum Lösen und einer Zusammenfassung der nötigen Theorie

Balken - Flächenträgheitsmomente | Aufgabe mit Lösung

Aufgabenstellung:

Lösungsweg:

Lösung: