Collapsed Shape of Shallow Unlined Tunnels Based on Functional Catastrophe Theory

<table class="figure-group"><tr class="fig-image" id="a"><td><object data="https://static.hindawi.com/articles/mpe/volume-2015/681257/figures/681257.fig.004a.svgz" name="681257.fig.004a" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(a) </b>Variations of <svg height="9.0231pt" id="M160" version="1.1" viewbox="0 -9.0231 9.29406 9.0231" width="9.29406pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M686 28C612 35 607 44 591 112C563 234 541 360 519 489L489 666L457 658L147 121C100 40 89 36 24 28L17 0H240L250 28C168 34 159 41 190 101L262 237H482C495 180 503 137 510 91C517 47 514 35 441 28L433 0H677L686 28ZM475 280H285L429 541H431L475 280Z" id="g113-66"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="703" vert-adv-y="703"></glyph.data></g></svg></td></tr><tr class="fig-image" id="b"><td><object data="https://static.hindawi.com/articles/mpe/volume-2015/681257/figures/681257.fig.004b.svgz" name="681257.fig.004b" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(b) </b>Variations of <svg height="9.60567pt" id="M161" style="vertical-align:-3.53608pt" version="1.1" viewbox="0 -6.06959 6.99087 9.60567" width="6.99087pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M516 277C516 378 460 448 376 448C335 448 288 434 240 401C158 345 127 277 107 159C53 -161 38 -221 23 -247L28 -261C43 -260 95 -250 114 -226C122 -216 129 -199 146 -75C152 -29 157 0 164 31C182 9 206 -12 237 -12C267 -12 325 8 376 46C451 102 516 179 516 277ZM432 259C432 134 366 40 259 40C236 40 204 48 171 76L195 203C219 327 253 400 326 400C398 400 432 336 432 259Z" id="g113-239"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="539" vert-adv-y="539"></glyph.data></g></svg></td></tr><tr class="fig-image" id="c"><td><object data="https://static.hindawi.com/articles/mpe/volume-2015/681257/figures/681257.fig.004c.svgz" name="681257.fig.004c" type="image/svg+xml"></object></td></tr><tr class="fig-caption"><td><b>(c) </b>Variations of <svg height="9.73165pt" id="M162" style="vertical-align:-3.37755pt" version="1.1" viewbox="0 -6.3541 10.7153 9.73165" width="10.7153pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M548 455L523 469C505 448 490 439 461 439C416 439 374 447 316 447C144 447 23 310 23 161C23 47 89 -12 179 -12C312 -12 429 106 429 244C429 303 419 344 368 386L371 388C404 383 440 378 474 378C507 378 530 411 548 455ZM350 274C350 191 308 25 198 25C144 25 108 75 108 157C108 264 170 395 276 395C296 395 320 384 333 360C347 334 350 316 350 274Z" id="g113-240"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="565" vert-adv-y="565"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,7.045,3.264)"><path d="M387 400C387 425 348 451 303 451C247 451 176 414 132 376C69 322 24 228 24 148C24 43 74 -12 147 -12C211 -12 301 33 363 103L346 128C319 99 249 51 193 51C148 51 112 84 112 165C112 230 130 287 154 330C170 359 199 400 243 400C277 400 304 383 326 354C333 345 343 343 354 348C378 360 387 382 387 400Z" id="g50-100"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="411" vert-adv-y="411"></glyph.data></g></svg> and <svg height="9.73165pt" id="M163" style="vertical-align:-3.37755pt" version="1.1" viewbox="0 -6.3541 10.1652 9.73165" width="10.1652pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M548 455L523 469C505 448 490 439 461 439C416 439 374 447 316 447C144 447 23 310 23 161C23 47 89 -12 179 -12C312 -12 429 106 429 244C429 303 419 344 368 386L371 388C404 383 440 378 474 378C507 378 530 411 548 455ZM350 274C350 191 308 25 198 25C144 25 108 75 108 157C108 264 170 395 276 395C296 395 320 384 333 360C347 334 350 316 350 274Z" id="g113-240"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="565" vert-adv-y="565"></glyph.data></g><g transform="matrix(.0095,0,0,-0.0095,7.045,3.264)"><path d="M329 433H203L239 587L230 596L147 534L123 433H57L30 395L34 388H115L61 129C37 16 59 -12 85 -12C147 -12 222 58 260 98L241 125C212 95 160 62 144 62C132 62 127 71 138 126L192 386L305 394L329 433Z" id="g50-117"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="347" vert-adv-y="347"></glyph.data></g></svg></td></tr></table>

<div>Shapes of the collapsing blocks versus rock mass parameters.</div>

Mathematical Problems in Engineering

fig4

Figure 4

Figure 4: Collapsed Shape of Shallow Unlined Tunnels Based on Functional Catastrophe Theory 