Analysis of Tapered Timoshenko and Euler–Bernoulli Beams on an Elastic Foundation with Moving Loads

<table class="table-group" id="tab7"><tr><td><table class="table"><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr><td class="align_left"><span style="width: 56.6773ptpx;"><svg height="14.5707pt" id="M141" style="vertical-align:-5.28237pt" version="1.1" viewbox="-0.0498162 -9.28833 56.6773 14.5707" width="56.6773pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M699 442H515V410C577 401 582 389 573 351C563 308 532 208 517 161H515C476 254 437 345 401 437H362C326 342 293 250 258 163H256C237 225 217 294 197 358C186 392 192 401 250 410V442H3V410C45 405 55 395 71 347C108 235 151 112 191 -11H229C267 84 302 175 340 268C377 177 418 83 457 -11H495C545 129 587 249 623 342C644 394 651 402 699 410V442Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.912,3.132)"><path d="M465 45L455 77C444 72 433 69 421 69C404 69 389 81 389 128V293C389 349 380 392 337 425C310 445 280 454 239 454C182 443 111 409 73 383C51 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transform="matrix(.013,0,0,-0.013,29.012,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,33.51,0)"><path d="M823 0V32C771 40 767 43 767 100V293C767 395 719 454 632 454C603 454 576 441 554 426C530 409 507 394 482 376C464 421 424 454 368 454C339 454 312 442 281 422C259 409 240 395 218 379V456C157 439 89 425 42 418V389C89 382 93 373 93 315V105C93 42 88 39 30 32V0H265V32C227 38 220 43 220 100V344C248 366 273 376 297 376C334 376 366 354 366 282V100C366 41 362 38 315 32V0H546V32C501 38 494 43 494 100V302C494 317 493 331 492 342C524 369 549 376 569 376C601 376 639 359 639 277V105C639 42 632 38 588 32V0H823Z"></path></g><g transform="matrix(.013,0,0,-0.013,44.51,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg></span></td><td class="align_center">Error <svg height="11.5564pt" id="M143" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 19.0986 11.5564" width="19.0986pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path d="M601 624L563 650L155 -11L192 -37L601 624ZM197 635C88 635 41 536 41 460C41 385 88 285 196 285C307 285 352 385 352 461C352 536 305 635 197 635ZM197 599C246 599 254 511 254 461C254 415 244 321 198 321C151 321 139 413 139 462C139 509 151 599 197 599ZM571 338C461 338 414 239 414 162C414 87 461 -12 570 -12C680 -12 725 87 725 163S678 338 571 338ZM569 302C618 302 627 214 627 164C627 118 618 24 571 24C523 24 512 116 512 164C512 216 523 302 569 302Z"></path></g><g transform="matrix(.013,0,0,-0.013,14.467,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg></td></tr><tr><td class="align_left" colspan="3"><hr/></td></tr><tr><td class="align_left">3.4930<svg height="6.01072pt" id="M144" style="vertical-align:-0.04980993pt" version="1.1" viewbox="-0.0498162 -5.96091 7.75925 6.01072" width="7.75925pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M471 153C471 170 463 194 452 212C400 220 373 229 322 255C373 281 400 290 452 298C463 316 471 339 471 357C456 366 431 371 410 370C377 329 356 310 308 279C311 336 317 364 336 413C326 432 310 451 294 459C279 451 262 432 252 413C271 364 277 336 280 279C232 310 211 329 178 370C157 371 132 367 117 357C117 340 125 316 136 298C188 290 215 281 266 255C215 229 188 220 136 212C125 194 117 171 117 153C132 144 157 139 178 140C211 181 232 200 280 231C277 174 271 146 252 97C262 78 278 59 294 51C309 59 326 78 336 97C317 146 311 174 308 231C356 200 377 181 410 140C431 139 456 143 471 153Z"></path></g></svg>10–6</td><td class="align_center">3.4877<svg height="6.01072pt" id="M145" style="vertical-align:-0.04980993pt" version="1.1" viewbox="-0.0498162 -5.96091 7.75925 6.01072" width="7.75925pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M471 153C471 170 463 194 452 212C400 220 373 229 322 255C373 281 400 290 452 298C463 316 471 339 471 357C456 366 431 371 410 370C377 329 356 310 308 279C311 336 317 364 336 413C326 432 310 451 294 459C279 451 262 432 252 413C271 364 277 336 280 279C232 310 211 329 178 370C157 371 132 367 117 357C117 340 125 316 136 298C188 290 215 281 266 255C215 229 188 220 136 212C125 194 117 171 117 153C132 144 157 139 178 140C211 181 232 200 280 231C277 174 271 146 252 97C262 78 278 59 294 51C309 59 326 78 336 97C317 146 311 174 308 231C356 200 377 181 410 140C431 139 456 143 471 153Z"></path></g></svg>10–6</td><td class="align_center">0.15</td></tr><tr class="table-tr"><td colspan="3"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Deflection of S-S beam (1), (<i>F</i> = 30 N).</div>

Journal of Mathematics

tab7

Table 7

Table 7: Analysis of Tapered Timoshenko and Euler–Bernoulli Beams on an Elastic Foundation with Moving Loads 