Querkraftschub

schubspannung

Biegelinie

Flächenträgheitsmomente

Normalspannungen

Produkte

Cauchy

Differentialrechnung 

Lineare Abbildungen

Taylorreihe/Taylorpolynom

Konvergenz rekursiver nichtmonotoner Zahlenfolgen

dualraum

Jordan Normal FOrm

Körperaxiomen

intr

leerlaufspannung

Integrale 

Totale Differenzierbarkeit

Mittelwertsatz

Mathe I

euklidischer Algorithmus

Brückenschaltung

Lineare Algebra 

QR-Zerlegung

Cholesky-Zerlegung

Stützmpment

Simplex Algorithmus

Optimierung in mehreren Variablen + Envelop Theorem 

Exterma Untersuchung

Basiswechselmatrix

Differenzenverstärker

übertragungsfunktion

Darstellungsmatrizen in neuer Basis darstellen

Basis wechsel, Transformationsmatrizen

Monotieverhalten

Cauchy-Produktformel

physik

elastischer stoss

lösbarkeit einer Matrix

kinematik

impuls

kugelkoordinaten une zylinderkoordinaten 

Matrix

Integralgleichung

kettenregel

Lösbarkeit einer Matrix

Glechgewicht

Schiefe Ebene

Cauchy Verdichtungssatz 

Schaltvorgänge

Singulärwertzerlegung

Spann

Lineares Gleichungssystem

spannung

Basis bestimmen

Fourier-Reihen

Unterschied In/Homogen

Gruppen

lineare abbildung 

quadratische ergänzung

basis 

Bild & Urbild

Zweigstromanalyse

Transistor als Thermometer

Investment

Teilfolgen

Epsilon Delta Kriterium

prinzip der virtuellen verrückung 

Linear Kombination

Cholesky Zerlegung

Polynomfunktionen beschreiben

Teilchenphysik und Festkörperphysik

test

Regel von de L'Hopital

KRAFT

Grenzwert Funktion 

Folgenkriterium

ϵ-δ Kriterium 

Beweis Stetigkeit 

Basiswechselsatz

Vorderrad fahradgabel lagerkräfte berechnen 

Spannungsregelung

Test

Induktion 

Jordan-Normalform

Inverse einer Matrix

Gleichmäßige Stetigkeit

addition

Stationäre Stoffübertragung

komplexe Ungleichungen 

Rekursive Folgen

Exponentialreihe

restglied

Newton Verfahren

Randwertaufgabe

Numerische Integration: Trapez- und Simpsonregel

dualräume

Dualraum

Vorlesung Technische Mechanik

<ol style="list-style-type: lower-alpha;">
<li>\( I_{y}=7 a^{3} t / 12\) <br />\(I_{z}=2 a^{3} t / 3\) <br />\(I_{y z}=-a^{3} t / 2\)<br /><br /></li>
<li>\(Etv_{B}=-200 F \)<br />\(Etv_{C}=-1525 \mathrm{~F} / 2\) <br />\(Etw_{B}=350 F\) <br />\(Etw_{C}=1225 F \)</li>
</ol>

<p>\begin{array}{|c|ccc|c|} \hline &amp; \text { Profil 1 } &amp; \text { Profil 2 } &amp; \text { Profil 3 } &amp; \text { Summe } \\ \hline A_{i} &amp; a t &amp; a t &amp; a t &amp; 3 a t \\ y_{S i} &amp; a / 2 &amp; 0 &amp; -a / 2 &amp; \\ z_{S i} &amp; a / 2 &amp; 0 &amp; -a / 2 &amp; \\ \hline I_{Y i} &amp; 0 &amp; a^{3} t / 12 &amp; 0 &amp; a^{3} t / 12 \\ I_{Z i} &amp; a^{3} t / 12 &amp; 0 &amp; a^{3} t / 12 &amp; a^{3} t / 6 \\ \hline z_{S i}^{2} A_{i} &amp; a^{3} t / 4 &amp; 0 &amp; a^{3} t / 4 &amp; a^{3} t / 2 \\ y_{S i}^{2} A_{i} &amp; a^{3} t / 4 &amp; 0 &amp; a^{3} t / 4 &amp; a^{3} t / 2 \\ -y_{S i} z_{S i} A_{i} &amp; -a^{3} t / 4 &amp; 0 &amp; -a^{3} t / 4 &amp; -a^{3} t / 2 \\ \hline \end{array}</p>


<p>\begin{align}<br />I_{y}&amp;=\frac{a^{3} t}{12}+\frac{a^{3} t}{2}=\frac{7}{12} a^{3} t \\\\<br />I_{z}&amp;=\frac{a^{3} t}{6}+\frac{a^{3} t}{2}=\frac{2}{3} a^{3} t \\\\<br />I_{y z}&amp;=-\frac{a^{3} t}{2} <br />\end{align}</p>


<p><strong>b) Verschiebungen</strong></p>


<p>Die Verschiebungen können mithilfe von Superposition ermittelt werden.</p>


<p>Für beide Lastfälle benötigte Werte:</p>


<p>\[<br />\begin{align}<br />1-\frac{I_{y z}^{2}}{I_{y} I_{z}}&amp;=1-\frac{a^{6} t^{2} \cdot 12 \cdot 3}{4 \cdot 7 \cdot 2 a^{6} t^{2}}=1-\frac{9}{14}=\frac{5}{14} \\\\<br />\frac{I_{y z}}{I_{y}}&amp;=-\frac{1}{2} \cdot \frac{12}{7}=-\frac{6}{7} \\ \\<br />\frac{I_{y z}}{I_{z}}&amp;=-\frac{1}{2} \cdot \frac{3}{2}=-\frac{3}{4}<br />\end{align}<br />\]</p>


<p>Lastfall 1: Kraft bei \( x_{B} \)</p>


<p>Lastfall 1: Kraft bei <mjx-container class="MathJax" jax="SVG"><svg alt=" x_{B} " style="vertical-align: -0.339ex" xmlns="http://www.w3.org/2000/svg" width="2.696ex" height="1.339ex" role="img" focusable="false" viewBox="0 -442 1191.7 592" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-764-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-764-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-764-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D435" xlink:href="#MJX-764-TEX-I-1D435"></use></g></g></g></g></g></svg></mjx-container></p>


<p>\begin{align}<br />\sum F_{y}=0&amp;:-A_{y}+F=0 \\\\</p>
<p>&amp;\Rightarrow A_{y}=F <br />\end{align}</p>


<p>\begin{align}<br />\sum M_{z}^{A}=0 &amp;:-M_{A}+5 a F=0 \\\\<br />&amp;\Rightarrow M_{A}=5 a F <br />\end{align}</p>


<p>\[<br />Q_{1 y}=-\left(-A_{y}+\left\langle x-\left.5 a\right\rangle^{0} F\right)=F\left(1-\langle x-5 a\rangle^{0}\right)\right.<br />\]</p>
<p>\[<br />M_{1 z}=-\left(-M_{A}+A_{y} x-\langle x-5 a\rangle F\right)=F(5 a-x+\langle x-5 a\rangle)<br />\]</p>


<p>\[<br />\bar{M}_{1 y}=\frac{-M_{1 z} I_{y z} / I_{z}}{1-I_{yz}^{2} /\left(I_{y} I_{z}\right)}=\frac{3}{4} \cdot \frac{14}{5} M_{1 z}=\frac{21}{10} M_{1 z}=\frac{21}{10} F a\left(5-\frac{x}{a}+\left\langle\frac{x}{a}-5\right\rangle\right)<br />\]</p>
<p>\[<br />\bar{M}_{1 z}=\frac{M_{1 z}}{1-I_{y z}^{2} /\left(I_{y} I_{z}\right)}=\frac{14}{5} M_{1 z}=\frac{14}{5} F a\left(5-\frac{x}{a}+\left\langle\frac{x}{a}-5\right\rangle\right)<br />\]</p>


<p>\begin{align}<br />E I_{z} \frac{d^{2} v_{1}}{d x^{2}}&amp;=\bar{M}_{1 z}=\frac{14}{5} F a\left(5-\frac{x}{a}+\left\langle\frac{x}{a}-5\right\rangle\right)\\\\<br />E I_{z} \frac{d v_{1}}{d x}&amp;=\frac{14}{5} F a^{2}\left(5 \frac{x}{a}-\frac{1}{2} \frac{x^{2}}{a^{2}}+\frac{1}{2}\left\langle\frac{x}{a}-5\right\rangle^{2}+c_{1}\right) \\\\<br />E I_{z} v_{1}&amp;=\frac{14}{5} F a^{3}\left(\frac{5}{2} \frac{x^{2}}{a^{2}}-\frac{1}{6} \frac{x^{3}}{a^{3}}+\frac{1}{6}\left\langle\frac{x}{a}-5\right\rangle^{3}+c_{1} x+c_{2}\right)<br />\end{align}</p>


<p>\begin{align}<br />v_{1}(0)&amp;=0 \quad \Rightarrow c_{2}=0 \\\\<br />\frac{d v_{1}}{d x}(0)&amp;=0 \quad \Rightarrow c_{1}=0<br />\end{align}</p>


<p>\[<br />\begin{aligned}<br />v_{1}(x) &amp;=\frac{14}{30} \frac{F a^{3}}{E I_{z}}\left[15\left(\frac{x}{a}\right)^{2}-\left(\frac{x}{a}\right)^{3}+\left\langle\frac{x}{a}-5\right\rangle^{3}\right] \\\\<br />&amp;=\frac{14}{30} \cdot \frac{3}{2} \frac{F}{E t}\left[15\left(\frac{x}{a}\right)^{2}-\left(\frac{x}{a}\right)^{3}+\left\langle\frac{x}{a}-5\right\rangle^{3}\right] \\\\<br />&amp;=\frac{7}{10} \frac{F}{E t}\left[15\left(\frac{x}{a}\right)^{2}-\left(\frac{x}{a}\right)^{3}+\left\langle\frac{x}{a}-5\right\rangle^{3}\right]<br />\end{aligned}<br />\]</p>


<p>\[<br />\begin{aligned}<br />&amp;v_{1 B}=v_{1}(5 a)=\frac{7}{10} \frac{F}{E t}\left(15 \cdot 5^{2}-5^{3}\right)=\frac{7 \cdot 25 F}{10 E t}(15-5)=175 \frac{F}{E t} \\\\<br />&amp;v_{1 C}=v_{1}(10 a)=\frac{7}{10} \frac{F}{E t}\left(15 \cdot 10^{2}-10^{3}+5^{3}\right)=\frac{875 F}{2 E t}<br />\end{aligned}<br />\]</p>


<p>Die Randbedingungen für die Vertikalverschiebung \( w(x) \) stimmen mit den Randbedingungen für die Horizontalverschiebung \( v(x) \) überein.</p>


<p>Die Randbedingungen für die Vertikalverschiebung <mjx-container class="MathJax" jax="SVG"><svg alt=" w(x) " style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="4.674ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2066 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-775-TEX-I-1D464" d="M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z"></path><path id="MJX-775-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-775-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-775-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></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="1D464" xlink:href="#MJX-775-TEX-I-1D464"></use></g><g data-mml-node="mo" transform="translate(716,0)"><use data-c="28" xlink:href="#MJX-775-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(1105,0)"><use data-c="1D465" xlink:href="#MJX-775-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1677,0)"><use data-c="29" xlink:href="#MJX-775-TEX-N-29"></use></g></g></g></svg></mjx-container> stimmen mit den Randbedingungen für die Horizontalverschiebung <mjx-container class="MathJax" jax="SVG"><svg alt=" v(x) " style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="4.152ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1835 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-776-TEX-I-1D463" d="M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 284 21 287Q21 294 29 316T53 368T97 419T160 441Q202 441 225 417T249 361Q249 344 246 335Q246 329 231 291T200 202T182 113Q182 86 187 69Q200 26 250 26Q287 26 319 60T369 139T398 222T409 277Q409 300 401 317T383 343T365 361T357 383Q357 405 376 424T417 443Q436 443 451 425T467 367Q467 340 455 284T418 159T347 40T241 -11Q177 -11 139 22Q102 54 102 117Q102 148 110 181T151 298Q173 362 173 380Z"></path><path id="MJX-776-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-776-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-776-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></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="1D463" xlink:href="#MJX-776-TEX-I-1D463"></use></g><g data-mml-node="mo" transform="translate(485,0)"><use data-c="28" xlink:href="#MJX-776-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(874,0)"><use data-c="1D465" xlink:href="#MJX-776-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1446,0)"><use data-c="29" xlink:href="#MJX-776-TEX-N-29"></use></g></g></g></svg></mjx-container> überein.</p>


<p>\begin{align}<br />E I_{y} \frac{d^{2} w_{1}}{d x^{2}}&amp;=-\bar{M}_{1 y}=-\frac{21}{10} F a\left(5-\frac{x}{a}+\left\langle\frac{x}{a}-5\right\rangle\right) \\\\<br />E I_{y} w_{1} &amp;=-\frac{21}{10} F a^{3}\left(\frac{5}{2} \frac{x^{2}}{a^{2}}-\frac{1}{6} \frac{x^{3}}{a^{3}}+\frac{1}{6}\left\langle\frac{x}{a}-5\right\rangle^{3}\right)<br />\end{align}\]</p>


<p>\begin{aligned} <br />w_{1}(x) &amp;=-\frac{21}{60} \frac{F a^{3}}{E I_{y}}\left[15\left(\frac{x}{a}\right)^{2}-\left(\frac{x}{a}\right)^{3}+\left\langle\frac{x}{a}-5\right\rangle^{3}\right] \\ <br />&amp;=-\frac{21}{60} \cdot \frac{12}{7} \frac{F}{E t}\left[15\left(\frac{x}{a}\right)^{2}-\left(\frac{x}{a}\right)^{3}+\left\langle\frac{x}{a}-5\right\rangle^{3}\right] \\ <br />&amp;=-\frac{3}{5} \frac{F}{E t}\left[15\left(\frac{x}{a}\right)^{2}-\left\langle\frac{x}{a}\right)^{3}+\left\langle\frac{x}{a}-5\right\rangle^{3}\right] \\\\<br />w_{1 B}=&amp;-\frac{3}{5} \frac{F}{E t}\left(15 \cdot 5^{2}-5^{3}\right)=-150 \frac{F}{E t} \\ <br />w_{1 C}=&amp;-\frac{3}{5} \frac{F}{E t}\left(15 \cdot 10^{2}-10^{3}+5^{3}\right)=-375 \frac{F}{E t} <br />\end{aligned}</p>


<p>Lastfall 2: Kraft bei \( x_{C} \)</p>


<p>Lastfall 2: Kraft bei <mjx-container class="MathJax" jax="SVG"><svg alt=" x_{C} " style="vertical-align: -0.375ex" xmlns="http://www.w3.org/2000/svg" width="2.698ex" height="1.375ex" role="img" focusable="false" viewBox="0 -442 1192.4 607.6" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-779-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-779-TEX-I-1D436" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-779-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D436" xlink:href="#MJX-779-TEX-I-1D436"></use></g></g></g></g></g></svg></mjx-container></p>


<p>\[<br />\begin{align}<br />\sum F_{z}=0:&amp; \quad -A_{z}+F=0 \\<br />&amp;\Rightarrow A_{z}=F \\\\<br />\sum M_{y}^{A}=0:&amp; \quad M_{A}-10 a F=0 \\\<br />&amp;\Rightarrow M_{A}=10 a F<br />\end{align}<br />\]</p>


<p>\begin{align}<br />Q_{2 z}=-\left(-A_{z}\right)=A_{z}=F <br />\end{align}</p>


<p>\begin{align}<br />M_{2 y}=-\left(M_{A}-A_{z} x\right)=-F(10 a-x <br />\end{align}</p>


<p>\[\begin{align} \bar{M}_{2 y}&amp;=\frac{M_{1 y}}{1-I_{y z}^{2} /\left(I_{y} I_{z}\right)}=-\frac{14}{5} F a\left(10-\frac{x}{a}\right) \\\\<br />\bar{M}_{2 z}&amp;=\frac{-M_{1 y} I_{y z} / I_{y}}{1-I_{y z}^{2} /\left(I_{y} I_{z}\right)}=\frac{14}{5}\left(\frac{-6}{7}\right) F a\left(10-\frac{x}{a}\right)=-\frac{12}{5} F a\left(10-\frac{x}{a}\right) \end{align}\]</p>


<p>\begin{align} <br />E I_{z} \frac{d^{2} v_{2}}{d x^{2}}&amp;=\bar{M}_{2 z}=-\frac{12}{5} F a\left(10-\frac{x}{a}\right) \\\\<br />E I_{z} \frac{d v_{2}}{d x}&amp;=-\frac{12}{5} F a^{2}\left(10 \frac{x}{a}-\frac{1}{2} \frac{x^{2}}{a^{2}}+d_{1}\right) \\\\<br />E I_{z} v_{2}&amp;=-\frac{12}{5} F a^{3}\left(5 \frac{x^{2}}{a^{2}}-\frac{1}{6} \frac{x^{3}}{a^{3}}+d_{1} x+d_{2}\right) <br />\end{align}</p>


<p>\begin{align}<br />v_{2}(0)=0 \quad \Rightarrow d_{2}=0 \\<br />\frac{d v_{2}}{d x}(0)=0 \quad \Rightarrow d_{1}=0 <br />\end{align}</p>


<p>\begin{align}<br />v_{2}(x)&amp;=-\frac{2}{5} \frac{F a^{3}}{E I_{z}}\left[30\left(\frac{x}{a}\right)^{2}-\left(\frac{x}{a}\right)^{3}\right]=-\frac{3}{5} \frac{F}{E t}\left[30\left(\frac{x}{a}\right)^{2}-\left(\frac{x}{a}\right)^{3}\right] \\\\<br />v_{2 B}&amp;=v_{2}(5 a)=-\frac{3}{5} \frac{F}{E t}\left(30 \cdot 5^{2}-5^{3}\right)=-\frac{3 \cdot 625 F}{5 E t}=-375 \frac{F}{E t} \\\\<br />v_{2 C}&amp;=v_{2}(10 a)=-\frac{3}{5} \frac{F}{E t}\left(30 \cdot 10^{2}-10^{3}\right)=-\frac{3 \cdot 2000 F}{5 E t}=-1200 \frac{F}{E t} <br />\end{align}</p>


<p>\begin{align}<br />E I_{y} \frac{d^{2} w_{2}}{d x^{2}}&amp;=-\bar{M}_{2 y}=\frac{14}{5} F a\left(10-\frac{x}{a}\right)\\\\<br />E I_{y} w_{2}&amp;=\frac{14}{5} F a^{3}\left(5 \frac{x^{2}}{a^{2}}-\frac{1}{6} \frac{x^{3}}{a^{3}}\right)\\\\<br />w_{2}&amp;=\frac{14}{30} \frac{F a^{3}}{E I_{y}}\left[30\left(\frac{x}{a}\right)^{2}-\left(\frac{x}{a}\right)^{3}\right]=\frac{4}{5} \frac{F}{E t}\left[30\left(\frac{x}{a}\right)^{2}-\left(\frac{x}{a}\right)^{3}\right] <br />\end{align}</p>


<p>\begin{align}<br />w_{2 B}=w_{2}(5 a)=\frac{4 \cdot 625}{5} \frac{F}{E t}=500 \frac{F}{E t} <br />\end{align}</p>


<p>\begin{align}<br />w_{2 C}=w_{2}(10 a)=\frac{4 \cdot 2000}{5} \frac{F}{E t}=1600 \frac{F}{E t} <br />\end{align}</p>


<p>\begin{align}<br />v_{B}=v_{1 B}+v_{2 B}=(175-375) \frac{F}{E t}=-200 \frac{F}{E t} <br />\end{align}</p>


<p>\begin{align}<br />w_{B}=w_{1 B}+w_{2 B}=(-150+500) \frac{F}{E t}=350 \frac{F}{E t} <br />\end{align}</p>


<p>\begin{align}<br />v_{C}=v_{1 C}+v_{2 C}=\left(\frac{875}{2}-1200\right) \frac{F}{E t}=-\frac{1525}{2} \frac{F}{E t} <br />\end{align}</p>


<p>\begin{align}<br />w_{C}=w_{1 C}+w_{2 C}=(-375+1600) \frac{F}{E t}=1225 \frac{F}{E t} <br />\end{align}</p>

<div class="horizontalLayoutContainer" style="display: flex; flex-direction: row;">
<div class="horizontalLayoutElement" style="padding-right: 10px; flex: 1;">
<p>Der abgebildete dünnwandige Balken (Wandstärke \( t \), Elastizitätsmodul \( E \) ) ist am Ende \( A \) fest eingespannt und wird an den \( F \) belastet.</p>
<ol style="list-style-type: lower-alpha;">
<li>Bestimmen Sie die Flächenträgheitsmomente im eingezeichneten Koordinatensystem</li>
<li>Bestimmen Sie die Verschiebungen an den Stellen \( x_{B} \) und \( x_{C} \).</li>
</ol>
</div>
<div class="horizontalLayoutElement" style="padding-left: 10px; flex: 1;">
<p class="horizontalLayoutText"><img style="width: 100%;" src="https://storage.googleapis.com/max-academy-image-bucket/6030a13a99dd3d088221a025.png" alt="Abbildung" /></p>
</div>
</div>
<p> </p>

<div class="horizontalLayoutContainer" style="display: flex; flex-direction: row;">
<div class="horizontalLayoutElement" style="padding-right: 10px; flex: 1;">
<p>Der abgebildete dünnwandige Balken (Wandstärke <mjx-container class="MathJax" jax="SVG"><svg alt=" t " style="vertical-align: -0.025ex" xmlns="http://www.w3.org/2000/svg" width="0.817ex" height="1.441ex" role="img" focusable="false" viewBox="0 -626 361 637" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-755-TEX-I-1D461" d="M26 385Q19 392 19 395Q19 399 22 411T27 425Q29 430 36 430T87 431H140L159 511Q162 522 166 540T173 566T179 586T187 603T197 615T211 624T229 626Q247 625 254 615T261 596Q261 589 252 549T232 470L222 433Q222 431 272 431H323Q330 424 330 420Q330 398 317 385H210L174 240Q135 80 135 68Q135 26 162 26Q197 26 230 60T283 144Q285 150 288 151T303 153H307Q322 153 322 145Q322 142 319 133Q314 117 301 95T267 48T216 6T155 -11Q125 -11 98 4T59 56Q57 64 57 83V101L92 241Q127 382 128 383Q128 385 77 385H26Z"></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="1D461" xlink:href="#MJX-755-TEX-I-1D461"></use></g></g></g></svg></mjx-container>, Elastizitätsmodul <mjx-container class="MathJax" jax="SVG"><svg alt=" E " style="vertical-align: 0" xmlns="http://www.w3.org/2000/svg" width="1.729ex" height="1.538ex" role="img" focusable="false" viewBox="0 -680 764 680" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-756-TEX-I-1D438" d="M492 213Q472 213 472 226Q472 230 477 250T482 285Q482 316 461 323T364 330H312Q311 328 277 192T243 52Q243 48 254 48T334 46Q428 46 458 48T518 61Q567 77 599 117T670 248Q680 270 683 272Q690 274 698 274Q718 274 718 261Q613 7 608 2Q605 0 322 0H133Q31 0 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H757Q764 676 764 669Q764 664 751 557T737 447Q735 440 717 440H705Q698 445 698 453L701 476Q704 500 704 528Q704 558 697 578T678 609T643 625T596 632T532 634H485Q397 633 392 631Q388 629 386 622Q385 619 355 499T324 377Q347 376 372 376H398Q464 376 489 391T534 472Q538 488 540 490T557 493Q562 493 565 493T570 492T572 491T574 487T577 483L544 351Q511 218 508 216Q505 213 492 213Z"></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="1D438" xlink:href="#MJX-756-TEX-I-1D438"></use></g></g></g></svg></mjx-container> ) ist am Ende <mjx-container class="MathJax" jax="SVG"><svg alt=" A " style="vertical-align: 0" xmlns="http://www.w3.org/2000/svg" width="1.697ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 750 716" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-757-TEX-I-1D434" d="M208 74Q208 50 254 46Q272 46 272 35Q272 34 270 22Q267 8 264 4T251 0Q249 0 239 0T205 1T141 2Q70 2 50 0H42Q35 7 35 11Q37 38 48 46H62Q132 49 164 96Q170 102 345 401T523 704Q530 716 547 716H555H572Q578 707 578 706L606 383Q634 60 636 57Q641 46 701 46Q726 46 726 36Q726 34 723 22Q720 7 718 4T704 0Q701 0 690 0T651 1T578 2Q484 2 455 0H443Q437 6 437 9T439 27Q443 40 445 43L449 46H469Q523 49 533 63L521 213H283L249 155Q208 86 208 74ZM516 260Q516 271 504 416T490 562L463 519Q447 492 400 412L310 260L413 259Q516 259 516 260Z"></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="1D434" xlink:href="#MJX-757-TEX-I-1D434"></use></g></g></g></svg></mjx-container> fest eingespannt und wird an den <mjx-container class="MathJax" jax="SVG"><svg alt=" F " style="vertical-align: 0" xmlns="http://www.w3.org/2000/svg" width="1.695ex" height="1.538ex" role="img" focusable="false" viewBox="0 -680 749 680" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-758-TEX-I-1D439" d="M48 1Q31 1 31 11Q31 13 34 25Q38 41 42 43T65 46Q92 46 125 49Q139 52 144 61Q146 66 215 342T285 622Q285 629 281 629Q273 632 228 634H197Q191 640 191 642T193 659Q197 676 203 680H742Q749 676 749 669Q749 664 736 557T722 447Q720 440 702 440H690Q683 445 683 453Q683 454 686 477T689 530Q689 560 682 579T663 610T626 626T575 633T503 634H480Q398 633 393 631Q388 629 386 623Q385 622 352 492L320 363H375Q378 363 398 363T426 364T448 367T472 374T489 386Q502 398 511 419T524 457T529 475Q532 480 548 480H560Q567 475 567 470Q567 467 536 339T502 207Q500 200 482 200H470Q463 206 463 212Q463 215 468 234T473 274Q473 303 453 310T364 317H309L277 190Q245 66 245 60Q245 46 334 46H359Q365 40 365 39T363 19Q359 6 353 0H336Q295 2 185 2Q120 2 86 2T48 1Z"></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="1D439" xlink:href="#MJX-758-TEX-I-1D439"></use></g></g></g></svg></mjx-container> belastet.</p>
<ol style="list-style-type: lower-alpha;">
<li>Bestimmen Sie die Flächenträgheitsmomente im eingezeichneten Koordinatensystem</li>
<li>Bestimmen Sie die Verschiebungen an den Stellen <mjx-container class="MathJax" jax="SVG"><svg alt=" x_{B} " style="vertical-align: -0.339ex" xmlns="http://www.w3.org/2000/svg" width="2.696ex" height="1.339ex" role="img" focusable="false" viewBox="0 -442 1191.7 592" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-759-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-759-TEX-I-1D435" d="M231 637Q204 637 199 638T194 649Q194 676 205 682Q206 683 335 683Q594 683 608 681Q671 671 713 636T756 544Q756 480 698 429T565 360L555 357Q619 348 660 311T702 219Q702 146 630 78T453 1Q446 0 242 0Q42 0 39 2Q35 5 35 10Q35 17 37 24Q42 43 47 45Q51 46 62 46H68Q95 46 128 49Q142 52 147 61Q150 65 219 339T288 628Q288 635 231 637ZM649 544Q649 574 634 600T585 634Q578 636 493 637Q473 637 451 637T416 636H403Q388 635 384 626Q382 622 352 506Q352 503 351 500L320 374H401Q482 374 494 376Q554 386 601 434T649 544ZM595 229Q595 273 572 302T512 336Q506 337 429 337Q311 337 310 336Q310 334 293 263T258 122L240 52Q240 48 252 48T333 46Q422 46 429 47Q491 54 543 105T595 229Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-759-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D435" xlink:href="#MJX-759-TEX-I-1D435"></use></g></g></g></g></g></svg></mjx-container> und <mjx-container class="MathJax" jax="SVG"><svg alt=" x_{C} " style="vertical-align: -0.375ex" xmlns="http://www.w3.org/2000/svg" width="2.698ex" height="1.375ex" role="img" focusable="false" viewBox="0 -442 1192.4 607.6" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-760-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-760-TEX-I-1D436" d="M50 252Q50 367 117 473T286 641T490 704Q580 704 633 653Q642 643 648 636T656 626L657 623Q660 623 684 649Q691 655 699 663T715 679T725 690L740 705H746Q760 705 760 698Q760 694 728 561Q692 422 692 421Q690 416 687 415T669 413H653Q647 419 647 422Q647 423 648 429T650 449T651 481Q651 552 619 605T510 659Q484 659 454 652T382 628T299 572T226 479Q194 422 175 346T156 222Q156 108 232 58Q280 24 350 24Q441 24 512 92T606 240Q610 253 612 255T628 257Q648 257 648 248Q648 243 647 239Q618 132 523 55T319 -22Q206 -22 128 53T50 252Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-760-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D436" xlink:href="#MJX-760-TEX-I-1D436"></use></g></g></g></g></g></svg></mjx-container>.</li>
</ol>
</div>
<div class="horizontalLayoutElement" style="padding-left: 10px; flex: 1;">
<p class="horizontalLayoutText"><img style="width: 100%;" src="https://storage.googleapis.com/max-academy-image-bucket/6030a13a99dd3d088221a025.png" alt="Abbildung" /></p>
</div>
</div>
<p> </p>

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

Aufgabenstellung:

Lösungsweg:

Lösung: