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<p>$<br />\sigma(x)=\frac{1}{x}\left(\frac{F_{0} \mathrm{I}}{A_{0}}-\frac{1}{2} \gamma a^{2}\right)+\frac{1}{2} \gamma x <br />$</p><p>$\mathrm{x}^{*}=\sqrt{\frac{\frac{\mathrm{F}_{0} \mathrm{I}}{\mathrm{A}_{0}}-\frac{1}{2} \gamma \mathrm{a}^{2}}{\frac{1}{2} \gamma}} <br />$</p><p>$<br />\sigma\left(x^{*}\right)=\frac{1}{x^{*}}\left(\frac{F_{0} \mathrm{I}}{A_{0}}-\frac{1}{2} \gamma a^{2}\right)+\frac{1}{2} \gamma x^{*} <br />$</p><p>$<br />\Delta \mathrm{I}=\frac{1}{\mathrm{E}}\left(\ln \left(\frac{\mathrm{a}}{\mathrm{l}}\right)\left(-\frac{1}{2} \gamma \mathrm{a}^{2}+\frac{\mathrm{F}_{0} \mathrm{I}}{\mathrm{A}_{0}}\right)+\frac{1}{4} \gamma\left(\mathrm{a}^{2}-\mathrm{I}^{2}\right)\right) <br />$</p>

<p>Mit $ g(x)=\gamma A(x)=\gamma \frac{A_{0}}{\mathrm{I} } x $ folgt der Normalkraftverlauf</p>

<p>Mit <mjx-container class="MathJax" jax="SVG"><svg alt=" g(x)=\gamma A(x)=\gamma \frac{A_{0}}{\mathrm{I} } x " style="vertical-align: -0.781ex" xmlns="http://www.w3.org/2000/svg" width="21.564ex" height="2.96ex" role="img" focusable="false" viewBox="0 -963.4 9531.1 1308.4" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-170-TEX-I-1D454" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 405Q262 405 221 352Q191 313 171 233T151 117Q151 38 213 38Q269 38 323 108L331 118L384 328Z"></path><path id="MJX-170-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-170-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-170-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><path id="MJX-170-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-170-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-170-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><path id="MJX-170-TEX-N-30" d="M96 585Q152 666 249 666Q297 666 345 640T423 548Q460 465 460 320Q460 165 417 83Q397 41 362 16T301 -15T250 -22Q224 -22 198 -16T137 16T82 83Q39 165 39 320Q39 494 96 585ZM321 597Q291 629 250 629Q208 629 178 597Q153 571 145 525T137 333Q137 175 145 125T181 46Q209 16 250 16Q290 16 318 46Q347 76 354 130T362 333Q362 478 354 524T321 597Z"></path><path id="MJX-170-TEX-N-49" d="M328 0Q307 3 180 3T32 0H21V46H43Q92 46 106 49T126 60Q128 63 128 342Q128 620 126 623Q122 628 118 630T96 635T43 637H21V683H32Q53 680 180 680T328 683H339V637H317Q268 637 254 634T234 623Q232 620 232 342Q232 63 234 60Q238 55 242 53T264 48T317 46H339V0H328Z"></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="1D454" xlink:href="#MJX-170-TEX-I-1D454"></use></g><g data-mml-node="mo" transform="translate(477,0)"><use data-c="28" xlink:href="#MJX-170-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(866,0)"><use data-c="1D465" xlink:href="#MJX-170-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1438,0)"><use data-c="29" xlink:href="#MJX-170-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(2104.8,0)"><use data-c="3D" xlink:href="#MJX-170-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(3160.6,0)"><use data-c="1D6FE" xlink:href="#MJX-170-TEX-I-1D6FE"></use></g><g data-mml-node="mi" transform="translate(3703.6,0)"><use data-c="1D434" xlink:href="#MJX-170-TEX-I-1D434"></use></g><g data-mml-node="mo" transform="translate(4453.6,0)"><use data-c="28" xlink:href="#MJX-170-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(4842.6,0)"><use data-c="1D465" xlink:href="#MJX-170-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(5414.6,0)"><use data-c="29" xlink:href="#MJX-170-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(6081.3,0)"><use data-c="3D" xlink:href="#MJX-170-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(7137.1,0)"><use data-c="1D6FE" xlink:href="#MJX-170-TEX-I-1D6FE"></use></g><g data-mml-node="mfrac" transform="translate(7680.1,0)"><g data-mml-node="msub" transform="translate(220,457.1) scale(0.707)"><g data-mml-node="mi"><use data-c="1D434" xlink:href="#MJX-170-TEX-I-1D434"></use></g><g data-mml-node="TeXAtom" transform="translate(783,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-170-TEX-N-30"></use></g></g></g><g data-mml-node="TeXAtom" transform="translate(511.9,-345) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="49" xlink:href="#MJX-170-TEX-N-49"></use></g></g><rect width="1039" height="60" x="120" y="220"></rect></g><g data-mml-node="mi" transform="translate(8959.1,0)"><use data-c="1D465" xlink:href="#MJX-170-TEX-I-1D465"></use></g></g></g></svg></mjx-container> folgt der Normalkraftverlauf</p>

<p>\[<br />N(x)=\int_{\xi=x} g(\xi) d \xi=\gamma \frac{A_{0}}{\mathrm{I}} \int_{\xi=x} \xi d \xi=\frac{1}{2} \gamma \frac{A_{0}}{\mathrm{I}} x^{2}+C_{1} <br />\]</p>

<p>\[<br />\sigma(x)=\frac{N(x)}{A(x)} <br />\]</p>

<p>Eine statische Randbedingung zur Bestimmung der Konstanten $\mathrm{C}_{1}$ nutzen:</p>

<p>Eine statische Randbedingung zur Bestimmung der Konstanten <mjx-container class="MathJax" jax="SVG"><svg alt="\mathrm{C}_{1}" style="vertical-align: -0.339ex" xmlns="http://www.w3.org/2000/svg" width="2.621ex" height="1.934ex" role="img" focusable="false" viewBox="0 -705 1158.6 855" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-173-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><path id="MJX-173-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></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="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mi"><use data-c="43" xlink:href="#MJX-173-TEX-N-43"></use></g></g><g data-mml-node="TeXAtom" transform="translate(755,-150) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-173-TEX-N-31"></use></g></g></g></g></g></svg></mjx-container> nutzen:</p>

<p>\[<br />\mathrm{N}(\mathrm{x}=\mathrm{a})=\mathrm{F}_{0}=\frac{1}{2} \gamma \frac{\mathrm{A}_{0}}{\mathrm{I}} \mathrm{a}^{2}+\mathrm{C}_{1}\]</p>

<p>\[\Rightarrow \quad \mathrm{C}_{1}=\mathrm{F}_{0}-\frac{1}{2} \gamma \frac{\mathrm{A}_{0}}{\mathrm{I}} \mathrm{a}^{2} <br />\]</p>

<p>Daraus folgt der Normalkraftverlauf</p>

<p>\[<br />N(x)=\frac{1}{2} \gamma \frac{A_{0}}{\mathrm{I} }\left(x^{2}-a^{2}\right)+F_{0} <br />\]</p>

<p>\[<br />\sigma(x)=\frac{1}{2} \gamma \frac{A_{0}}{\mathrm{I}} \frac{\mathrm{I}}{A_{0}} \frac{1}{x}\left(x^{2}-a^{2}\right)+\frac{F_{0} \mathrm{I}}{A_{0} x}=\frac{1}{x}\left(\frac{F_{0} \mathrm{I}}{A_{0}}-\frac{1}{2} \gamma a^{2}\right)+\frac{1}{2} \gamma x <br />\]</p>

<p>Der Ort und der Betrag der kleinsten Spannung $ \sigma\left(x^{*}\right) $ ergibt sich aus der ersten Variation nach $ x $</p>

<p>Der Ort und der Betrag der kleinsten Spannung <mjx-container class="MathJax" jax="SVG"><svg alt=" \sigma\left(x^{*}\right) " style="vertical-align: -0.566ex" xmlns="http://www.w3.org/2000/svg" width="5.711ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2524.2 1000" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-178-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-178-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-178-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-178-TEX-N-2217" d="M229 286Q216 420 216 436Q216 454 240 464Q241 464 245 464T251 465Q263 464 273 456T283 436Q283 419 277 356T270 286L328 328Q384 369 389 372T399 375Q412 375 423 365T435 338Q435 325 425 315Q420 312 357 282T289 250L355 219L425 184Q434 175 434 161Q434 146 425 136T401 125Q393 125 383 131T328 171L270 213Q283 79 283 63Q283 53 276 44T250 35Q231 35 224 44T216 63Q216 80 222 143T229 213L171 171Q115 130 110 127Q106 124 100 124Q87 124 76 134T64 161Q64 166 64 169T67 175T72 181T81 188T94 195T113 204T138 215T170 230T210 250L74 315Q65 324 65 338Q65 353 74 363T98 374Q106 374 116 368T171 328L229 286Z"></path><path id="MJX-178-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-178-TEX-I-1D70E"></use></g><g data-mml-node="mrow" transform="translate(737.7,0)"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-178-TEX-N-28"></use></g><g data-mml-node="msup" transform="translate(389,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-178-TEX-I-1D465"></use></g><g data-mml-node="TeXAtom" transform="translate(605,363) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="2217" xlink:href="#MJX-178-TEX-N-2217"></use></g></g></g><g data-mml-node="mo" transform="translate(1397.6,0)"><use data-c="29" xlink:href="#MJX-178-TEX-N-29"></use></g></g></g></g></svg></mjx-container> ergibt sich aus der ersten Variation nach <mjx-container class="MathJax" jax="SVG"><svg alt=" x " style="vertical-align: -0.025ex" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-179-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></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="1D465" xlink:href="#MJX-179-TEX-I-1D465"></use></g></g></g></svg></mjx-container></p>

<p>\[<br />\frac{\mathrm{d} \sigma\left(\mathrm{x}=\mathrm{x}^{*}\right)}{\mathrm{dx}}=-\frac{1}{\mathrm{x}^{2}}\left(\frac{\mathrm{F}_{0} \mathrm{I}}{\mathrm{A}_{0}}-\frac{1}{2} \gamma \mathrm{a}^{2}\right)+\frac{1}{2} \gamma=0 <br />\]</p>

<p>Daraus läßt sich der Ort bestimmen</p>

<p>\[<br />\frac{\mathrm{F}_{0} \mathrm{I}}{\mathrm{A}_{0}}-\frac{1}{2} \gamma \mathrm{a}^{2}=\frac{1}{2} \gamma \mathrm{x}^{2}\]</p><p>\[\Rightarrow \quad \mathrm{x}^{*}=\sqrt{\frac{\frac{\mathrm{F}_{0} \mathrm{I}}{\mathrm{A}_{0}}-\frac{1}{2} \gamma \mathrm{a}^{2}}{\frac{1}{2} \gamma}} <br />\]</p>

<p>Durch die zweite Variation wird das Minimum bestätigt</p>

<p>\[<br />\frac{\mathrm{d}^{2} \sigma\left(\mathrm{x}=\mathrm{x}^{*}\right)}{\mathrm{dx}^{2}}=-(-2) \frac{1}{\mathrm{x}^{* 3}}\left(\frac{\mathrm{F}_{0} \mathrm{I}}{\mathrm{A}_{0}}-\frac{1}{2} \gamma \mathrm{a}^{2}\right)=2 \frac{1}{\mathrm{x}^{* 3}}\left(\frac{\mathrm{F}_{0} \mathrm{I}}{\mathrm{A}_{0}}-\frac{1}{2} \gamma \mathrm{a}^{2}\right) &gt; 0 <br />\]</p><p>Für $ \mathrm{x}^{*} &gt; 0 $ und $ \frac{\mathrm{F}_{0} \mathrm{I}}{\mathrm{A}_{0}} &gt; \frac{1}{2} \gamma \mathrm{a}^{2} $ existiert ein Spannungsminimum.</p>

<p>Damit ist der Betrag der kleinsten Normalspannung</p>

<p>\[<br />\sigma\left(x^{*}\right)=\frac{1}{x^{*}}\left(\frac{F_{0} \mathrm{I}}{A_{0}}-\frac{1}{2} \gamma a^{2}\right)+\frac{1}{2} \gamma x^{*} <br />\]</p>

<p>Die Gesamtverlängerung $ \Delta \mathrm{I} $ des Stabes ergibt sich aus der Integration der Differentialgleichung </p>

<p>Die Gesamtverlängerung <mjx-container class="MathJax" jax="SVG"><svg alt=" \Delta \mathrm{I} " style="vertical-align: 0" xmlns="http://www.w3.org/2000/svg" width="2.701ex" height="1.62ex" role="img" focusable="false" viewBox="0 -716 1194 716" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-187-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-187-TEX-N-49" d="M328 0Q307 3 180 3T32 0H21V46H43Q92 46 106 49T126 60Q128 63 128 342Q128 620 126 623Q122 628 118 630T96 635T43 637H21V683H32Q53 680 180 680T328 683H339V637H317Q268 637 254 634T234 623Q232 620 232 342Q232 63 234 60Q238 55 242 53T264 48T317 46H339V0H328Z"></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-187-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="49" xlink:href="#MJX-187-TEX-N-49"></use></g></g></g></g></svg></mjx-container> des Stabes ergibt sich aus der Integration der Differentialgleichung </p>

<p>\begin{align}<br />\frac{d u(x)}{d x}&amp;=\frac{N(x)}{E A(x)}=\frac{1}{E}\left(\frac{1}{2} \gamma \frac{A_{0}}{\mathrm{I}} \frac{\mathrm{I}}{A_{0}} \frac{1}{x}\left(x^{2}-a^{2}\right)+\frac{F_{0} \mathrm{I}}{A_{0} x}\right) <br />\\&amp;=\frac{1}{E}\left(\frac{1}{x}\left(-\frac{1}{2} \gamma a^{2}+\frac{F_{0} \mathrm{I}}{A_{0}}\right)+\frac{1}{2} \gamma x\right) <br />\end{align}</p>

<p>\[<br />u(x)=\frac{1}{E}\left(\ln (x)\left(-\frac{1}{2} \gamma a^{2}+\frac{F_{0} \mathrm{I}}{A_{0}}\right)+\frac{1}{4} \gamma x^{2}\right)+C_{2} <br />\]</p>

<p>Die geometrische Randbedingung zur Bestimmung der Konstanten lautet</p>

<p>\[<br />\mathrm{u}(\mathrm{x}=\mathrm{I})=0=\frac{1}{\mathrm{E}}\left(\ln (\mathrm{I})\left(-\frac{1}{2} \gamma \mathrm{a}^{2}+\frac{\mathrm{F}_{0} \mathrm{I}}{\mathrm{A}_{0}}\right)+\frac{1}{4} \gamma \mathrm{I}^{2}\right)+\mathrm{C}_{2} <br />\]</p>

<p>\[<br />\Rightarrow \quad \mathrm{C}_{2}=-\frac{1}{\mathrm{E}}\left(\ln (\mathrm{I})\left(-\frac{1}{2} \gamma \mathrm{a}^{2}+\frac{\mathrm{F}_{0} \mathrm{I}}{\mathrm{A}_{0}}\right)+\frac{1}{4} \gamma\mathrm{I}^{2}\right) <br />\]</p>

<p>Damit lautet der Verschiebungsverlauf</p>

<p>\[ \mathrm{u}(\mathrm{x})=\frac{1}{\mathrm{E}}\left(\ln (\mathrm{x})\left(-\frac{1}{2} \gamma \mathrm{a}^{2}+\frac{\mathrm{F}_{0} \mathrm{I}}{\mathrm{A}_{0}}\right)+\frac{1}{4} \gamma \mathrm{x}^{2}-\left(\ln (\mathrm{I})\left(-\frac{1}{2} \gamma \mathrm{a}^{2}+\frac{\mathrm{F}_{0} \mathrm{I}}{\mathrm{A}_{0}}\right)+\frac{1}{4} \gamma \mathrm{I}^{2}\right)\right. \]</p>

<p>mit $ \ln (x)-\ln (I)=\ln \left(\frac{x}{1}\right) $ folgt</p>

<p>mit <mjx-container class="MathJax" jax="SVG"><svg alt=" \ln (x)-\ln (I)=\ln \left(\frac{x}{1}\right) " style="vertical-align: -0.791ex" xmlns="http://www.w3.org/2000/svg" width="21.758ex" height="2.713ex" role="img" focusable="false" viewBox="0 -849.5 9617.1 1199" xmlns:xlink="http://www.w3.org/1999/xlink"><defs><path id="MJX-193-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 690T171 693T182 694H185V379Q185 62 186 60Q190 52 198 49Q219 46 247 46H263V0H255L232 1Q209 2 183 2T145 3T107 3T57 1L34 0H26V46H42Z"></path><path id="MJX-193-TEX-N-6E" d="M41 46H55Q94 46 102 60V68Q102 77 102 91T102 122T103 161T103 203Q103 234 103 269T102 328V351Q99 370 88 376T43 385H25V408Q25 431 27 431L37 432Q47 433 65 434T102 436Q119 437 138 438T167 441T178 442H181V402Q181 364 182 364T187 369T199 384T218 402T247 421T285 437Q305 442 336 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xlink:href="#MJX-193-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(2406.2,0)"><use data-c="2212" xlink:href="#MJX-193-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(3406.4,0)"><use data-c="6C" xlink:href="#MJX-193-TEX-N-6C"></use><use data-c="6E" xlink:href="#MJX-193-TEX-N-6E" transform="translate(278,0)"></use></g><g data-mml-node="mo" transform="translate(4240.4,0)"><use data-c="2061" xlink:href="#MJX-193-TEX-N-2061"></use></g><g data-mml-node="mo" transform="translate(4240.4,0)"><use data-c="28" xlink:href="#MJX-193-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(4629.4,0)"><use data-c="1D43C" xlink:href="#MJX-193-TEX-I-1D43C"></use></g><g data-mml-node="mo" transform="translate(5133.4,0)"><use data-c="29" xlink:href="#MJX-193-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(5800.2,0)"><use data-c="3D" xlink:href="#MJX-193-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(6856,0)"><use data-c="6C" xlink:href="#MJX-193-TEX-N-6C"></use><use data-c="6E" xlink:href="#MJX-193-TEX-N-6E" transform="translate(278,0)"></use></g><g data-mml-node="mo" transform="translate(7690,0)"><use data-c="2061" xlink:href="#MJX-193-TEX-N-2061"></use></g><g data-mml-node="mrow" transform="translate(7856.7,0)"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="28" xlink:href="#MJX-193-TEX-SO-28"></use></g><g data-mml-node="mfrac" transform="translate(458,0)"><g data-mml-node="mi" transform="translate(220,394) scale(0.707)"><use data-c="1D465" xlink:href="#MJX-193-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(245.5,-345) scale(0.707)"><use data-c="31" xlink:href="#MJX-193-TEX-N-31"></use></g><rect width="604.5" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(1302.5,0) translate(0 -0.5)"><use data-c="29" xlink:href="#MJX-193-TEX-SO-29"></use></g></g></g></g></svg></mjx-container> folgt</p>

<p>\[<br />\mathrm{u}(\mathrm{x})=\frac{1}{\mathrm{E}}\left(\ln \left(\frac{\mathrm{x}}{\mathrm{l}}\right)\left(-\frac{1}{2} \gamma \mathrm{a}^{2}+\frac{\mathrm{F}_{0} \mathrm{I}}{\mathrm{A}_{0}}\right)+\frac{1}{4} \gamma\left(\mathrm{x}^{2}-\mathrm{I}^{2}\right)\right) <br />\]</p><p>Die Verschiebung ist negativ in positiver $ x $ - Richtung.</p>

<p><mjx-container class="MathJax" jax="SVG" display="true" justify="left"><svg alt="
\mathrm{u}(\mathrm{x})=\frac{1}{\mathrm{E}}\left(\ln \left(\frac{\mathrm{x}}{\mathrm{l}}\right)\left(-\frac{1}{2} \gamma \mathrm{a}^{2}+\frac{\mathrm{F}_{0} \mathrm{I}}{\mathrm{A}_{0}}\right)+\frac{1}{4} \gamma\left(\mathrm{x}^{2}-\mathrm{I}^{2}\right)\right) 
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<p>\[<br />\Delta \mathrm{l}=\mathrm{u}(\mathrm{x}=\mathrm{a})=\frac{1}{\mathrm{E}}\left(\ln \left(\frac{\mathrm{a}}{\mathrm{l}}\right)\left(-\frac{1}{2} \gamma \mathrm{a}^{2}+\frac{\mathrm{F}_{0} \mathrm{I}}{\mathrm{A}_{0}}\right)+\frac{1}{4} \gamma\left(\mathrm{a}^{2}-\mathrm{I}^{2}\right)\right) . <br />\]</p>

<div class="horizontalLayoutContainer" style="display: flex; flex-direction: row;"><div class="horizontalLayoutElement" style="padding-right: 10px; flex: 2;"><p>Ein homogener Stab konstanter Dicke mit linear veränderlichem Querschnitt wird mit Eigengewicht belastet.</p><p> </p><p><strong>Gesucht:<br /></strong>Bestimmung des Zugspannungsverlaufs $ \sigma(\mathrm{x}) $ und des Orts $ \left(\mathrm{x}^{*}\right) $ und Betrags der kleinsten Spannung sowie der Gesamtverlängerung $ \Delta \mathrm{I} $</p><p> </p><p><strong>Gegeben:</strong> $ \mathrm{I},\ \mathrm{a},\ \mathrm{A}_{0},\ \rho,\ \mathrm{g},\ \mathrm{F}_{0},\ \mathrm{E} $</p></div><div class="horizontalLayoutElement" style="padding-left: 10px; flex: 1;"><p class="horizontalLayoutText"><img src="https://storage.googleapis.com/max-academy-image-bucket/608fb3e673a35faa3d16bf47.png" alt="Homogenen Stab konstanter Dicke mit linear veränderlichem Querschnitt" width="213" height="153" /></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 homogener Stab konstanter Dicke mit linear veränderlichem Querschnitt wird mit Eigengewicht belastet.</p><p> </p><p><strong>Gesucht:<br /></strong>Bestimmung des Zugspannungsverlaufs <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-166-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 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Dies ist eine interaktive Aufgabe zu: TM 2 Sammlung mit praktischen Tipps zum Lösen und einer Zusammenfassung der nötigen Theorie

Aufgabenstellung:

Lösungsweg:

Lösung: