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<p>\[<br />\sigma(\mathrm{x})=\gamma\left(\frac{1}{2} \mathrm{I}- \mathrm{x} \right)-\alpha_{T} E \Delta T\]</p><p>\[u=\frac{1}{2} \gamma \frac{1}{\mathrm{E}} \times(\mathrm{I}-\mathrm{x})\]</p><p> </p>

<p><strong>1. Lösungsmöglichkeit</strong></p>

<p>Die Differentialgleichung mit \( \gamma=\rho \mathrm{g} \) und \( \Delta \mathrm{T}= \) const. ergibt durch Integration den Verschiebungsverlauf</p>

<p>Die Differentialgleichung mit <mjx-container class="MathJax" jax="SVG"><svg alt=" \gamma=\rho \mathrm{g} " style="vertical-align: -0.489ex" xmlns="http://www.w3.org/2000/svg" width="6.547ex" height="1.808ex" role="img" focusable="false" viewBox="0 -583 2893.6 799" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-240-TEX-I-1D6FE" d="M31 249Q11 249 11 258Q11 275 26 304T66 365T129 418T206 441Q233 441 239 440Q287 429 318 386T371 255Q385 195 385 170Q385 166 386 166L398 193Q418 244 443 300T486 391T508 430Q510 431 524 431H537Q543 425 543 422Q543 418 522 378T463 251T391 71Q385 55 378 6T357 -100Q341 -165 330 -190T303 -216Q286 -216 286 -188Q286 -138 340 32L346 51L347 69Q348 79 348 100Q348 257 291 317Q251 355 196 355Q148 355 108 329T51 260Q49 251 47 251Q45 249 31 249Z"></path><path id="MJX-240-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-240-TEX-I-1D70C" d="M58 -216Q25 -216 23 -186Q23 -176 73 26T127 234Q143 289 182 341Q252 427 341 441Q343 441 349 441T359 442Q432 442 471 394T510 276Q510 219 486 165T425 74T345 13T266 -10H255H248Q197 -10 165 35L160 41L133 -71Q108 -168 104 -181T92 -202Q76 -216 58 -216ZM424 322Q424 359 407 382T357 405Q322 405 287 376T231 300Q217 269 193 170L176 102Q193 26 260 26Q298 26 334 62Q367 92 389 158T418 266T424 322Z"></path><path id="MJX-240-TEX-N-67" d="M329 409Q373 453 429 453Q459 453 472 434T485 396Q485 382 476 371T449 360Q416 360 412 390Q410 404 415 411Q415 412 416 414V415Q388 412 363 393Q355 388 355 386Q355 385 359 381T368 369T379 351T388 325T392 292Q392 230 343 187T222 143Q172 143 123 171Q112 153 112 133Q112 98 138 81Q147 75 155 75T227 73Q311 72 335 67Q396 58 431 26Q470 -13 470 -72Q470 -139 392 -175Q332 -206 250 -206Q167 -206 107 -175Q29 -140 29 -75Q29 -39 50 -15T92 18L103 24Q67 55 67 108Q67 155 96 193Q52 237 52 292Q52 355 102 398T223 442Q274 442 318 416L329 409ZM299 343Q294 371 273 387T221 404Q192 404 171 388T145 343Q142 326 142 292Q142 248 149 227T179 192Q196 182 222 182Q244 182 260 189T283 207T294 227T299 242Q302 258 302 292T299 343ZM403 -75Q403 -50 389 -34T348 -11T299 -2T245 0H218Q151 0 138 -6Q118 -15 107 -34T95 -74Q95 -84 101 -97T122 -127T170 -155T250 -167Q319 -167 361 -139T403 -75Z"></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="1D6FE" xlink:href="#MJX-240-TEX-I-1D6FE"></use></g><g data-mml-node="mo" transform="translate(820.8,0)"><use data-c="3D" xlink:href="#MJX-240-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(1876.6,0)"><use data-c="1D70C" xlink:href="#MJX-240-TEX-I-1D70C"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(2393.6,0)"><g data-mml-node="mi"><use data-c="67" xlink:href="#MJX-240-TEX-N-67"></use></g></g></g></g></svg></mjx-container> und <mjx-container class="MathJax" jax="SVG"><svg alt=" \Delta \mathrm{T}= " style="vertical-align: -0.186ex" xmlns="http://www.w3.org/2000/svg" width="5.907ex" height="1.805ex" role="img" focusable="false" viewBox="0 -716 2610.8 798" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-241-TEX-N-394" d="M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z"></path><path id="MJX-241-TEX-N-54" d="M36 443Q37 448 46 558T55 671V677H666V671Q667 666 676 556T685 443V437H645V443Q645 445 642 478T631 544T610 593Q593 614 555 625Q534 630 478 630H451H443Q417 630 414 618Q413 616 413 339V63Q420 53 439 50T528 46H558V0H545L361 3Q186 1 177 0H164V46H194Q264 46 283 49T309 63V339V550Q309 620 304 625T271 630H244H224Q154 630 119 601Q101 585 93 554T81 486T76 443V437H36V443Z"></path><path id="MJX-241-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></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="394" xlink:href="#MJX-241-TEX-N-394"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(833,0)"><g data-mml-node="mi"><use data-c="54" xlink:href="#MJX-241-TEX-N-54"></use></g></g><g data-mml-node="mo" transform="translate(1832.8,0)"><use data-c="3D" xlink:href="#MJX-241-TEX-N-3D"></use></g></g></g></svg></mjx-container> const. ergibt durch Integration den Verschiebungsverlauf</p>

<p>\[<br />\left(\mathrm{EA} \mathrm{u}^{\prime}\right)^{\prime}=\left(\alpha_{\mathrm{T}} \mathrm{EA} \Delta \mathrm{T}\right)^{\prime}-\gamma \mathrm{A} <br />\]</p>

<p>\[<br />E A u^{\prime}=\alpha_{T} E A \Delta T-\gamma A \mathrm{x}+C_{1} <br />\]</p>

<p>\[<br />\mathrm{EA} \mathrm{u}=\alpha_{T} \mathrm{E} \mathrm{A} \Delta \mathrm{T} \mathrm{x}-\frac{1}{2} \gamma \mathrm{A} \mathrm{x}^{2}+\mathrm{C}_{1} \mathrm{x}+\mathrm{C}_{2} <br />\]</p>

<p>Das System ist statisch unbestimmt. Es gibt nur geometrische Randbedingungen.<br />Die geometrische Randbedingung zur Bestimmung der Konstanten lautet</p>

<p>\[<br />\mathrm{u}(0)=0 \quad \Rightarrow \quad \mathrm{C}_{2}=0 <br />\]</p><p>\[<br />\mathrm{u}(\mathrm{I})=0 \quad\Rightarrow \quad \mathrm{C}_{1}=-\alpha_{\mathrm{T}} \mathrm{EA} \Delta \mathrm{T}+\frac{1}{2} \gamma \mathrm{A} \mathrm{I} <br />\]</p>

<p>Die Spannungs- und Verschiebungsverläufe mit \( \mathrm{N}=\mathrm{EA} \mathrm{u}^{\prime} \)</p>

<p>Die Spannungs- und Verschiebungsverläufe mit <mjx-container class="MathJax" jax="SVG"><svg alt=" \mathrm{N}=\mathrm{EA} \mathrm{u}^{\prime} " style="vertical-align: -0.186ex" xmlns="http://www.w3.org/2000/svg" width="9.837ex" height="1.903ex" role="img" focusable="false" viewBox="0 -759 4348 841" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-247-TEX-N-4E" d="M42 46Q74 48 94 56T118 69T128 86V634H124Q114 637 52 637H25V683H232L235 680Q237 679 322 554T493 303L578 178V598Q572 608 568 613T544 627T492 637H475V683H483Q498 680 600 680Q706 680 715 683H724V637H707Q634 633 622 598L621 302V6L614 0H600Q585 0 582 3T481 150T282 443T171 605V345L172 86Q183 50 257 46H274V0H265Q250 3 150 3Q48 3 33 0H25V46H42Z"></path><path id="MJX-247-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-247-TEX-N-45" d="M128 619Q121 626 117 628T101 631T58 634H25V680H597V676Q599 670 611 560T625 444V440H585V444Q584 447 582 465Q578 500 570 526T553 571T528 601T498 619T457 629T411 633T353 634Q266 634 251 633T233 622Q233 622 233 621Q232 619 232 497V376H286Q359 378 377 385Q413 401 416 469Q416 471 416 473V493H456V213H416V233Q415 268 408 288T383 317T349 328T297 330Q290 330 286 330H232V196V114Q232 57 237 52Q243 47 289 47H340H391Q428 47 452 50T505 62T552 92T584 146Q594 172 599 200T607 247T612 270V273H652V270Q651 267 632 137T610 3V0H25V46H58Q100 47 109 49T128 61V619Z"></path><path id="MJX-247-TEX-N-41" d="M255 0Q240 3 140 3Q48 3 39 0H32V46H47Q119 49 139 88Q140 91 192 245T295 553T348 708Q351 716 366 716H376Q396 715 400 709Q402 707 508 390L617 67Q624 54 636 51T687 46H717V0H708Q699 3 581 3Q458 3 437 0H427V46H440Q510 46 510 64Q510 66 486 138L462 209H229L209 150Q189 91 189 85Q189 72 209 59T259 46H264V0H255ZM447 255L345 557L244 256Q244 255 345 255H447Z"></path><path id="MJX-247-TEX-N-75" d="M383 58Q327 -10 256 -10H249Q124 -10 105 89Q104 96 103 226Q102 335 102 348T96 369Q86 385 36 385H25V408Q25 431 27 431L38 432Q48 433 67 434T105 436Q122 437 142 438T172 441T184 442H187V261Q188 77 190 64Q193 49 204 40Q224 26 264 26Q290 26 311 35T343 58T363 90T375 120T379 144Q379 145 379 161T380 201T380 248V315Q380 361 370 372T320 385H302V431Q304 431 378 436T457 442H464V264Q464 84 465 81Q468 61 479 55T524 46H542V0Q540 0 467 -5T390 -11H383V58Z"></path><path id="MJX-247-TEX-V-2032" d="M79 43Q73 43 52 49T30 61Q30 68 85 293T146 528Q161 560 198 560Q218 560 240 545T262 501Q262 496 260 486Q259 479 173 263T84 45T79 43Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="4E" xlink:href="#MJX-247-TEX-N-4E"></use></g></g><g data-mml-node="mo" transform="translate(1027.8,0)"><use data-c="3D" xlink:href="#MJX-247-TEX-N-3D"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(2083.6,0)"><g data-mml-node="mi"><use data-c="45" xlink:href="#MJX-247-TEX-N-45"></use><use data-c="41" xlink:href="#MJX-247-TEX-N-41" transform="translate(681,0)"></use></g></g><g data-mml-node="msup" transform="translate(3514.6,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="75" xlink:href="#MJX-247-TEX-N-75"></use></g></g><g data-mml-node="TeXAtom" transform="translate(589,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="2032" xlink:href="#MJX-247-TEX-V-2032"></use></g></g></g></g></g></svg></mjx-container></p>

<p>\[<br />\sigma(\mathrm{x})=\frac{1}{A} N=\frac{1}{A}\left(\frac{1}{2} \gamma A \mathrm{I}-\gamma A \mathrm{x} -\alpha_{T} E A \Delta T\right)=\gamma\left(\frac{1}{2} \mathrm{I}- \mathrm{x} \right)-\alpha_{T} E \Delta T <br />\]</p>

<p>\[<br />\mathrm{EA} \mathrm{u}=-\frac{1}{2} \gamma \mathrm{A} \mathrm{x}^{2}+\frac{1}{2} \gamma \mathrm{A} \mathrm{I} \mathrm{x} \quad \]</p>

<p>\[\Rightarrow \quad \mathrm{u}=\frac{1}{2} \gamma \frac{1}{\mathrm{E}} \times(\mathrm{I}-\mathrm{x}) <br />\]</p>

<p><strong>2. Lösungsmöglichkeit</strong></p>

<p>Die Bestimmung der Normalkraft- und Verschiebungsverläufe erfolgt über die Integration der Gleichungen:</p>

<p>\[<br />\frac{\mathrm{dN}}{\mathrm{dx}}=-\gamma \mathrm{A} \quad\]</p>

<p>\[\Rightarrow \quad \mathrm{N}=-\gamma \mathrm{A} \mathrm{x}+\mathrm{C}_{1}<br />\]</p>

<p>\[\frac{d u}{d \mathrm{x}}=\frac{N(x)}{E A}+\alpha \tau \Delta T=-\gamma \frac{1}{E} \mathrm{x}+\frac{C_{1}}{E A}+\alpha \tau \Delta T \]</p>

<p>\[\Rightarrow \quad u=-\frac{1}{2} \gamma \frac{1}{E} \mathrm{x}^{2}+\frac{C_{1}}{E A} \mathrm{x}+\alpha \tau \Delta T \mathrm{x}+C_{2}\]</p>

<p>Bestimmung der Integrationskonstanten aus den Randbedingungen und der Spannungs- und Verschiebungsverläufe wie oben</p>

<div class="horizontalLayoutContainer" style="display: flex; flex-direction: row;"><div class="horizontalLayoutElement" style="padding-right: 10px; flex: 2;"><p>Ein schwerer Stab (spezifisches Gewicht \( \gamma \), Querschnittsfläche A, Wärmeausdehnungskoeffitient \( \alpha_{T} \) ) unter einer gleichmäßigen Temperaturerwärmung um die Temperaturdifferenz \( \Delta \mathrm{T} \) ist in \( \mathrm{B} \) und in \( \mathrm{C} \) befestigt.</p><p> </p><p><strong>Gesucht:</strong> <br />Bestimmung der Spannungsverteilung \( \sigma(\mathrm{x}) \) und des Verschiebungsverlaufes \( \mathrm{u}(\mathrm{x}) \)</p><p> </p><p><strong>Gegeben: </strong>\( \gamma,\ \mathrm{A},\ \alpha_{\mathrm{T}},\ \mathrm{l},\ \Delta \mathrm{T} \)</p></div><div class="horizontalLayoutElement" style="padding-left: 10px; flex: 1;"><p class="horizontalLayoutText"><img src="https://storage.googleapis.com/max-academy-image-bucket/608fba7673a35faa3d16ce84.png" alt="Schwerer Stab unter einer gleichmäßigen Temperaturerwärmung" width="213" height="220" /></p></div></div><p> </p>

<div class="horizontalLayoutContainer" style="display: flex; flex-direction: row;"><div class="horizontalLayoutElement" style="padding-right: 10px; flex: 2;"><p>Ein schwerer Stab (spezifisches Gewicht <mjx-container class="MathJax" jax="SVG"><svg alt=" \gamma " style="vertical-align: -0.489ex" xmlns="http://www.w3.org/2000/svg" width="1.229ex" height="1.486ex" role="img" focusable="false" viewBox="0 -441 543 657" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-232-TEX-I-1D6FE" d="M31 249Q11 249 11 258Q11 275 26 304T66 365T129 418T206 441Q233 441 239 440Q287 429 318 386T371 255Q385 195 385 170Q385 166 386 166L398 193Q418 244 443 300T486 391T508 430Q510 431 524 431H537Q543 425 543 422Q543 418 522 378T463 251T391 71Q385 55 378 6T357 -100Q341 -165 330 -190T303 -216Q286 -216 286 -188Q286 -138 340 32L346 51L347 69Q348 79 348 100Q348 257 291 317Q251 355 196 355Q148 355 108 329T51 260Q49 251 47 251Q45 249 31 249Z"></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="1D6FE" xlink:href="#MJX-232-TEX-I-1D6FE"></use></g></g></g></svg></mjx-container>, Querschnittsfläche A, Wärmeausdehnungskoeffitient <mjx-container class="MathJax" jax="SVG"><svg alt=" \alpha_{T} " style="vertical-align: -0.339ex" xmlns="http://www.w3.org/2000/svg" width="2.762ex" height="1.339ex" role="img" focusable="false" viewBox="0 -442 1220.8 592" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-233-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><path id="MJX-233-TEX-I-1D447" d="M40 437Q21 437 21 445Q21 450 37 501T71 602L88 651Q93 669 101 677H569H659Q691 677 697 676T704 667Q704 661 687 553T668 444Q668 437 649 437Q640 437 637 437T631 442L629 445Q629 451 635 490T641 551Q641 586 628 604T573 629Q568 630 515 631Q469 631 457 630T439 622Q438 621 368 343T298 60Q298 48 386 46Q418 46 427 45T436 36Q436 31 433 22Q429 4 424 1L422 0Q419 0 415 0Q410 0 363 1T228 2Q99 2 64 0H49Q43 6 43 9T45 27Q49 40 55 46H83H94Q174 46 189 55Q190 56 191 56Q196 59 201 76T241 233Q258 301 269 344Q339 619 339 625Q339 630 310 630H279Q212 630 191 624Q146 614 121 583T67 467Q60 445 57 441T43 437H40Z"></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="1D6FC" xlink:href="#MJX-233-TEX-I-1D6FC"></use></g><g data-mml-node="TeXAtom" transform="translate(673,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="1D447" xlink:href="#MJX-233-TEX-I-1D447"></use></g></g></g></g></g></svg></mjx-container> ) unter einer gleichmäßigen Temperaturerwärmung um die Temperaturdifferenz <mjx-container class="MathJax" jax="SVG"><svg alt=" \Delta \mathrm{T} " style="vertical-align: 0" xmlns="http://www.w3.org/2000/svg" width="3.518ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 1555 716" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-234-TEX-N-394" d="M51 0Q46 4 46 7Q46 9 215 357T388 709Q391 716 416 716Q439 716 444 709Q447 705 616 357T786 7Q786 4 781 0H51ZM507 344L384 596L137 92L383 91H630Q630 93 507 344Z"></path><path id="MJX-234-TEX-N-54" d="M36 443Q37 448 46 558T55 671V677H666V671Q667 666 676 556T685 443V437H645V443Q645 445 642 478T631 544T610 593Q593 614 555 625Q534 630 478 630H451H443Q417 630 414 618Q413 616 413 339V63Q420 53 439 50T528 46H558V0H545L361 3Q186 1 177 0H164V46H194Q264 46 283 49T309 63V339V550Q309 620 304 625T271 630H244H224Q154 630 119 601Q101 585 93 554T81 486T76 443V437H36V443Z"></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="394" xlink:href="#MJX-234-TEX-N-394"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(833,0)"><g data-mml-node="mi"><use data-c="54" xlink:href="#MJX-234-TEX-N-54"></use></g></g></g></g></svg></mjx-container> ist in <mjx-container class="MathJax" jax="SVG"><svg alt=" \mathrm{B} " style="vertical-align: 0" xmlns="http://www.w3.org/2000/svg" width="1.602ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 708 683" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-235-TEX-N-42" d="M131 622Q124 629 120 631T104 634T61 637H28V683H229H267H346Q423 683 459 678T531 651Q574 627 599 590T624 512Q624 461 583 419T476 360L466 357Q539 348 595 302T651 187Q651 119 600 67T469 3Q456 1 242 0H28V46H61Q103 47 112 49T131 61V622ZM511 513Q511 560 485 594T416 636Q415 636 403 636T371 636T333 637Q266 637 251 636T232 628Q229 624 229 499V374H312L396 375L406 377Q410 378 417 380T442 393T474 417T499 456T511 513ZM537 188Q537 239 509 282T430 336L329 337H229V200V116Q229 57 234 52Q240 47 334 47H383Q425 47 443 53Q486 67 511 104T537 188Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="42" xlink:href="#MJX-235-TEX-N-42"></use></g></g></g></g></svg></mjx-container> und in <mjx-container class="MathJax" jax="SVG"><svg alt=" \mathrm{C} " style="vertical-align: -0.048ex" xmlns="http://www.w3.org/2000/svg" width="1.633ex" height="1.643ex" role="img" focusable="false" viewBox="0 -705 722 726" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-236-TEX-N-43" d="M56 342Q56 428 89 500T174 615T283 681T391 705Q394 705 400 705T408 704Q499 704 569 636L582 624L612 663Q639 700 643 704Q644 704 647 704T653 705H657Q660 705 666 699V419L660 413H626Q620 419 619 430Q610 512 571 572T476 651Q457 658 426 658Q322 658 252 588Q173 509 173 342Q173 221 211 151Q232 111 263 84T328 45T384 29T428 24Q517 24 571 93T626 244Q626 251 632 257H660L666 251V236Q661 133 590 56T403 -21Q262 -21 159 83T56 342Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="43" xlink:href="#MJX-236-TEX-N-43"></use></g></g></g></g></svg></mjx-container> befestigt.</p><p> </p><p><strong>Gesucht:</strong> <br />Bestimmung der Spannungsverteilung <mjx-container class="MathJax" jax="SVG"><svg alt=" \sigma(\mathrm{x}) " style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="4.247ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1877 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-237-TEX-I-1D70E" d="M184 -11Q116 -11 74 34T31 147Q31 247 104 333T274 430Q275 431 414 431H552Q553 430 555 429T559 427T562 425T565 422T567 420T569 416T570 412T571 407T572 401Q572 357 507 357Q500 357 490 357T476 358H416L421 348Q439 310 439 263Q439 153 359 71T184 -11ZM361 278Q361 358 276 358Q152 358 115 184Q114 180 114 178Q106 141 106 117Q106 67 131 47T188 26Q242 26 287 73Q316 103 334 153T356 233T361 278Z"></path><path id="MJX-237-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-237-TEX-N-78" d="M201 0Q189 3 102 3Q26 3 17 0H11V46H25Q48 47 67 52T96 61T121 78T139 96T160 122T180 150L226 210L168 288Q159 301 149 315T133 336T122 351T113 363T107 370T100 376T94 379T88 381T80 383Q74 383 44 385H16V431H23Q59 429 126 429Q219 429 229 431H237V385Q201 381 201 369Q201 367 211 353T239 315T268 274L272 270L297 304Q329 345 329 358Q329 364 327 369T322 376T317 380T310 384L307 385H302V431H309Q324 428 408 428Q487 428 493 431H499V385H492Q443 385 411 368Q394 360 377 341T312 257L296 236L358 151Q424 61 429 57T446 50Q464 46 499 46H516V0H510H502Q494 1 482 1T457 2T432 2T414 3Q403 3 377 3T327 1L304 0H295V46H298Q309 46 320 51T331 63Q331 65 291 120L250 175Q249 174 219 133T185 88Q181 83 181 74Q181 63 188 55T206 46Q208 46 208 23V0H201Z"></path><path id="MJX-237-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="1D70E" xlink:href="#MJX-237-TEX-I-1D70E"></use></g><g data-mml-node="mo" transform="translate(571,0)"><use data-c="28" xlink:href="#MJX-237-TEX-N-28"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(960,0)"><g data-mml-node="mi"><use data-c="78" xlink:href="#MJX-237-TEX-N-78"></use></g></g><g data-mml-node="mo" transform="translate(1488,0)"><use data-c="29" xlink:href="#MJX-237-TEX-N-29"></use></g></g></g></svg></mjx-container> und des Verschiebungsverlaufes <mjx-container class="MathJax" jax="SVG"><svg alt=" \mathrm{u}(\mathrm{x}) " style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="4.213ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1862 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-238-TEX-N-75" d="M383 58Q327 -10 256 -10H249Q124 -10 105 89Q104 96 103 226Q102 335 102 348T96 369Q86 385 36 385H25V408Q25 431 27 431L38 432Q48 433 67 434T105 436Q122 437 142 438T172 441T184 442H187V261Q188 77 190 64Q193 49 204 40Q224 26 264 26Q290 26 311 35T343 58T363 90T375 120T379 144Q379 145 379 161T380 201T380 248V315Q380 361 370 372T320 385H302V431Q304 431 378 436T457 442H464V264Q464 84 465 81Q468 61 479 55T524 46H542V0Q540 0 467 -5T390 -11H383V58Z"></path><path id="MJX-238-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-238-TEX-N-78" d="M201 0Q189 3 102 3Q26 3 17 0H11V46H25Q48 47 67 52T96 61T121 78T139 96T160 122T180 150L226 210L168 288Q159 301 149 315T133 336T122 351T113 363T107 370T100 376T94 379T88 381T80 383Q74 383 44 385H16V431H23Q59 429 126 429Q219 429 229 431H237V385Q201 381 201 369Q201 367 211 353T239 315T268 274L272 270L297 304Q329 345 329 358Q329 364 327 369T322 376T317 380T310 384L307 385H302V431H309Q324 428 408 428Q487 428 493 431H499V385H492Q443 385 411 368Q394 360 377 341T312 257L296 236L358 151Q424 61 429 57T446 50Q464 46 499 46H516V0H510H502Q494 1 482 1T457 2T432 2T414 3Q403 3 377 3T327 1L304 0H295V46H298Q309 46 320 51T331 63Q331 65 291 120L250 175Q249 174 219 133T185 88Q181 83 181 74Q181 63 188 55T206 46Q208 46 208 23V0H201Z"></path><path id="MJX-238-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="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="75" xlink:href="#MJX-238-TEX-N-75"></use></g></g><g data-mml-node="mo" transform="translate(556,0)"><use data-c="28" xlink:href="#MJX-238-TEX-N-28"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" 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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><path id="MJX-239-TEX-N-54" d="M36 443Q37 448 46 558T55 671V677H666V671Q667 666 676 556T685 443V437H645V443Q645 445 642 478T631 544T610 593Q593 614 555 625Q534 630 478 630H451H443Q417 630 414 618Q413 616 413 339V63Q420 53 439 50T528 46H558V0H545L361 3Q186 1 177 0H164V46H194Q264 46 283 49T309 63V339V550Q309 620 304 625T271 630H244H224Q154 630 119 601Q101 585 93 554T81 486T76 443V437H36V443Z"></path><path id="MJX-239-TEX-N-6C" d="M42 46H56Q95 46 103 60V68Q103 77 103 91T103 124T104 167T104 217T104 272T104 329Q104 366 104 407T104 482T104 542T103 586T103 603Q100 622 89 628T44 637H26V660Q26 683 28 683L38 684Q48 685 67 686T104 688Q121 689 141 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data-c="2C" xlink:href="#MJX-239-TEX-N-2C"></use></g><g data-mml-node="mtext" transform="translate(5333.2,0)"><use data-c="A0" xlink:href="#MJX-239-TEX-N-A0"></use></g><g data-mml-node="mi" transform="translate(5583.2,0)"><use data-c="394" xlink:href="#MJX-239-TEX-N-394"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(6416.2,0)"><g data-mml-node="mi"><use data-c="54" xlink:href="#MJX-239-TEX-N-54"></use></g></g></g></g></svg></mjx-container></p></div><div class="horizontalLayoutElement" style="padding-left: 10px; flex: 1;"><p class="horizontalLayoutText"><img src="https://storage.googleapis.com/max-academy-image-bucket/608fba7673a35faa3d16ce84.png" alt="Schwerer Stab unter einer gleichmäßigen Temperaturerwärmung" width="213" height="220" /></p></div></div><p> </p>

Dies ist eine interaktive Aufgabe zu: TM 2 Sammlung mit praktischen Tipps zum Lösen und einer Zusammenfassung der nötigen Theorie

Aufgabenstellung:

Lösungsweg:

Lösung: