Joint Effect of the Nonlinearity of Elastic Foundations and the Variation of the Inertia Ratio on Buckling Behavior of Prismatic and Nonprismatic Columns Using a GDQ Method

<table class="table-group" id="tab7"><tr><td><table class="table"><tr><td class="thead-hr" colspan="13"><hr/></td></tr><tr class="thead"><td class="align_left" rowspan="5">Boundary conditions</td><td class="align_center" colspan="12">Buckling load <svg height="11.5564pt" id="M140" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 16.3323 11.5564" width="16.3323pt" 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="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z"></path></g><g transform="matrix(.013,0,0,-0.013,11.672,0)"><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 class="thead"><td class="align_center" colspan="3">Present proposed technique (<i>N</i> = 15)</td><td class="align_center" colspan="3">Present proposed technique (<i>N</i> = 15)</td><td class="align_center" colspan="3">Present proposed technique (<i>N</i> = 15)</td><td class="align_center" colspan="3">Present proposed technique (<i>N</i> = 15)</td></tr><tr class="thead"><td class="align_center" colspan="3">Elastic foundation parameters</td><td class="align_center" colspan="3">Elastic foundation parameters</td><td class="align_center" colspan="3">Elastic foundation parameters</td><td class="align_center" colspan="3">Elastic foundation parameters</td></tr><tr class="thead"><td class="align_center"><i>K</i><sub>1</sub></td><td class="align_center"><i>K</i><sub>2</sub></td><td class="align_center"><i>K</i><sub>3</sub></td><td class="align_center"><i>K</i><sub>1</sub></td><td class="align_center"><i>K</i><sub>2</sub></td><td class="align_center"><i>K</i><sub>3</sub></td><td class="align_center"><i>K</i><sub>1</sub></td><td class="align_center"><i>K</i><sub>2</sub></td><td class="align_center"><i>K</i><sub>3</sub></td><td class="align_center"><i>K</i><sub>1</sub></td><td class="align_center"><i>K</i><sub>2</sub></td><td class="align_center"><i>K</i><sub>3</sub></td></tr><tr class="thead"><td class="align_center">0.0</td><td class="align_center">0.0</td><td class="align_center">0.0</td><td class="align_center">30</td><td class="align_center">0.0</td><td class="align_center">0.0</td><td class="align_center">30</td><td class="align_center">30</td><td class="align_center">0.0</td><td class="align_center">30</td><td class="align_center">30</td><td class="align_center">30</td></tr><tr><td class="thead-hr" colspan="13"><hr/></td></tr><tr><td class="align_left">Simple-simple</td><td class="align_center" colspan="3">20.7923</td><td class="align_center" colspan="3">23.7674</td><td class="align_center" colspan="3">53.7674</td><td class="align_center" colspan="3">64.0248</td></tr><tr><td class="align_left">Clamped-clamped</td><td class="align_center" colspan="3">81.9233</td><td class="align_center" colspan="3">84.1346</td><td class="align_center" colspan="3">114.1346</td><td class="align_center" colspan="3">114.1770</td></tr><tr><td class="align_left">Clamped-simple</td><td class="align_center" colspan="3">42.1091</td><td class="align_center" colspan="3">44.8215</td><td class="align_center" colspan="3">74.8215</td><td class="align_center" colspan="3">80.4827</td></tr><tr class="table-tr"><td colspan="13"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Effects of elastic foundation parameters <svg height="12.5794pt" id="M137" style="vertical-align:-3.29107pt" version="1.1" viewbox="-0.0498162 -9.28833 62.0449 12.5794" width="62.0449pt" 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="M290 -163V-135C183 -126 181 -122 181 -44V583C181 662 184 666 290 675V703H120V-163H290Z"></path></g><g transform="matrix(.013,0,0,-0.013,4.485,0)"><path d="M743 650H503L496 622L527 618C563 613 564 603 532 573C449 495 371 431 323 392C301 374 272 355 246 346L280 522C297 609 300 614 379 622L385 650H135L129 622C209 614 215 609 198 522L124 133C106 39 99 35 23 28L17 0H271L277 28C193 35 192 39 208 133L239 316C264 328 280 325 303 288C368 183 435 90 502 0H652L659 28C602 34 584 43 543 94C495 154 403 283 347 369L574 554C634 603 659 612 735 624L743 650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,13.715,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 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370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z"></path></g><g transform="matrix(.013,0,0,-0.013,57.301,0)"><path d="M226 -163V703H56V676C162 667 165 662 165 584V-43C165 -122 162 -126 56 -136V-163H226Z"></path></g></svg> on the nondimensional critical buckling loads <svg height="11.5564pt" id="M138" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 16.3323 11.5564" width="16.3323pt" 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="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 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Mathematical Problems in Engineering

tab7

Table 7

Table 7: Joint Effect of the Nonlinearity of Elastic Foundations and the Variation of the Inertia Ratio on Buckling Behavior of Prismatic and Nonprismatic Columns Using a GDQ Method 

Table 7 | Joint Effect of the Nonlinearity of Elastic Foundations and the Variation of the Inertia Ratio on Buckling Behavior of Prismatic and Nonprismatic Columns Using a GDQ Method 