Application of the Bipotential Theory to a Nonassociated Drucker–Prager Model

<div>Displacements of the tunnel with different dilatancy angles: (a) vertical displacement <svg height="14.1649pt" id="M153" style="vertical-align:-5.529pt" version="1.1" viewbox="-0.0498162 -8.6359 14.5977 14.1649" width="14.5977pt" 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="M715 650H476L470 622C550 619 562 610 562 570C562 536 553 481 535 384L523 321C485 121 408 37 283 37C178 37 99 109 134 297L178 532C193 612 199 616 278 622L284 650H29L23 622C105 616 109 609 94 532L48 291C23 160 51 83 98 40C143 -1 202 -16 260 -16C321 -16 387 3 439 49C509 111 542 204 563 312L577 384C595 479 608 533 619 569C629 603 637 620 709 622L715 650Z" id="g113-86"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.606,3.132)"><path d="M563 395C563 428 543 451 519 451C500 451 480 438 472 423C468 415 465 403 470 394C477 381 482 368 482 347C482 272 386 136 342 74H340C334 163 323 257 312 337C302 410 290 451 257 451C214 451 160 385 127 338L145 310C169 341 203 373 209 373C219 373 225 369 232 325C249 215 269 62 271 -15C220 -80 135 -165 18 -231L26 -257L142 -230C254 -105 279 -74 342 13C394 85 484 211 526 286C553 334 563 365 563 395Z" id="g50-122"></path></g></svg> at the top of the tunnel; (b) horizontal displacement <svg height="11.927pt" id="M154" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 14.4333 11.927" width="14.4333pt" 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="M715 650H476L470 622C550 619 562 610 562 570C562 536 553 481 535 384L523 321C485 121 408 37 283 37C178 37 99 109 134 297L178 532C193 612 199 616 278 622L284 650H29L23 622C105 616 109 609 94 532L48 291C23 160 51 83 98 40C143 -1 202 -16 260 -16C321 -16 387 3 439 49C509 111 542 204 563 312L577 384C595 479 608 533 619 569C629 603 637 620 709 622L715 650Z" id="g113-86"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.606,3.132)"><path d="M545 403C545 425 527 451 498 451C448 451 400 404 312 286L290 341C262 412 246 451 219 451C182 451 138 404 92 345L112 323C152 368 169 378 181 378C192 378 201 359 216 321L257 214C184 115 142 69 113 69C102 69 93 73 88 79S78 89 68 89C47 89 24 61 24 40C24 8 49 -12 75 -12C125 -12 175 31 271 175L312 64C330 15 357 -12 382 -12C421 -12 475 35 515 98L496 121C466 88 439 62 420 62C402 62 384 92 363 147L325 247C349 280 374 309 397 333C422 361 446 381 462 381C471 381 481 374 488 366C493 360 499 357 505 357C521 357 545 380 545 403Z" id="g50-121"></path></g></svg> at the bottom edge of the tunnel.</div>

Advances in Civil Engineering

fig13

Figure 13

Figure 13: Application of the Bipotential Theory to a Nonassociated Drucker–Prager Model 