Earth Retaining Walls CIVL455 CHAPTER II: LATERAL EARTH PRESSURE THEORY
Rankine’s Lateral Earth Pressure Theory
• Lateral pressure on walls can be evaluated for threeconditions;
At-rest condition Active condition
Passive condition
Ref: B. M. Das (2007)
At-rest lateral pressure
• Rankine’s lateral earth pressure theory based on a frictionlesswall and horizontal backfill. AT-REST (GEOSTATIC) CASE.
For most soils
For overconsolidated granular soils
For normally consolidated cohesive soils
For over-consolidated cohesive soils
Ref: B. M. Das (2007)
Active lateral pressure
• Rankine’s lateral earth pressure theory based on a frictionlesswall and horizontal backfill. ACTIVE CASE.
Drained
Undrained
Drained
DrainedUndrained
Undrained
Ref: B. M. Das (2007)
Active lateral pressure
• Rankine’s lateral earth pressure theory based on a frictionlesswall and inclined backfill. ACTIVE CASE.
Drained backfill with c’=0.
Drained backfill with c’>0.
Ref: B. M. Das (2007)
Passive lateral pressure
• Rankine’s lateral earth pressure theory based on a frictionlesswall and horizontal backfill. PASSIVE CASE.
Drained
Undrained
Passive lateral pressure
• Rankine’s lateral earth pressure theory based on a frictionlesswall and inclined backfill. PASSIVE CASE.
Drained backfill with c’=0.
Drained backfill with c’>0.
Ref: B. M. Das (2007)
Kp = Rankine's passive pressure coefficient
f () f
20 0.490 0.492 0.496 0.504 0.515 0.531 0.553 0.583 0.627 0.697 0.940 20 2.040 2.034 2.015 1.985 1.941 1.883 1.808 1.715 1.596 1.435 1.064
21 0.472 0.474 0.478 0.485 0.495 0.509 0.529 0.555 0.591 0.646 0.748 21 2.117 2.111 2.093 2.063 2.020 1.964 1.892 1.803 1.691 1.547 1.338
22 0.455 0.456 0.460 0.466 0.475 0.488 0.505 0.528 0.560 0.605 0.677 22 2.198 2.192 2.175 2.145 2.103 2.048 1.978 1.892 1.786 1.654 1.478
23 0.438 0.439 0.443 0.448 0.457 0.468 0.484 0.504 0.531 0.569 0.625 23 2.283 2.277 2.260 2.231 2.189 2.135 2.067 1.984 1.882 1.758 1.600
24 0.422 0.423 0.426 0.431 0.439 0.449 0.463 0.481 0.505 0.537 0.583 24 2.371 2.366 2.348 2.320 2.279 2.226 2.160 2.078 1.980 1.861 1.715
25 0.406 0.407 0.410 0.414 0.421 0.431 0.443 0.460 0.481 0.509 0.547 25 2.464 2.458 2.441 2.413 2.373 2.321 2.255 2.176 2.080 1.966 1.828
26 0.390 0.391 0.394 0.398 0.405 0.413 0.425 0.439 0.458 0.482 0.515 26 2.561 2.556 2.539 2.511 2.471 2.419 2.355 2.277 2.183 2.073 1.941
27 0.376 0.376 0.379 0.383 0.389 0.396 0.407 0.420 0.437 0.458 0.487 27 2.663 2.657 2.641 2.613 2.574 2.522 2.459 2.382 2.290 2.182 2.055
28 0.361 0.362 0.364 0.368 0.373 0.380 0.390 0.401 0.417 0.436 0.460 28 2.770 2.764 2.748 2.720 2.681 2.630 2.567 2.491 2.401 2.295 2.172
29 0.347 0.348 0.350 0.353 0.358 0.365 0.373 0.384 0.398 0.415 0.436 29 2.882 2.877 2.860 2.832 2.793 2.743 2.680 2.605 2.516 2.412 2.291
30 0.333 0.334 0.336 0.339 0.343 0.350 0.357 0.367 0.379 0.395 0.414 30 3.000 2.995 2.978 2.950 2.912 2.861 2.799 2.724 2.635 2.533 2.414
31 0.320 0.321 0.322 0.325 0.329 0.335 0.342 0.351 0.362 0.376 0.393 31 3.124 3.119 3.102 3.074 3.036 2.985 2.923 2.848 2.761 2.659 2.542
32 0.307 0.308 0.309 0.312 0.316 0.321 0.328 0.336 0.346 0.358 0.374 32 3.255 3.249 3.233 3.205 3.166 3.116 3.053 2.979 2.892 2.791 2.675
33 0.295 0.295 0.297 0.299 0.303 0.307 0.313 0.321 0.330 0.342 0.355 33 3.392 3.387 3.370 3.342 3.303 3.253 3.190 3.116 3.029 2.928 2.813
34 0.283 0.283 0.285 0.287 0.290 0.294 0.300 0.307 0.315 0.325 0.338 34 3.537 3.532 3.515 3.487 3.448 3.397 3.335 3.260 3.173 3.072 2.958
35 0.271 0.271 0.273 0.275 0.278 0.282 0.287 0.293 0.301 0.310 0.322 35 3.690 3.685 3.668 3.640 3.600 3.549 3.486 3.412 3.324 3.224 3.109
36 0.260 0.260 0.261 0.263 0.266 0.270 0.274 0.280 0.287 0.296 0.306 36 3.852 3.846 3.829 3.801 3.761 3.710 3.647 3.571 3.483 3.383 3.268
37 0.249 0.249 0.250 0.252 0.254 0.258 0.262 0.267 0.274 0.282 0.291 37 4.023 4.017 4.000 3.971 3.931 3.879 3.816 3.740 3.651 3.550 3.435
38 0.238 0.238 0.239 0.241 0.243 0.246 0.250 0.255 0.261 0.268 0.277 38 4.204 4.198 4.181 4.152 4.111 4.059 3.995 3.918 3.829 3.727 3.611
39 0.228 0.228 0.229 0.230 0.232 0.235 0.239 0.244 0.249 0.256 0.263 39 4.395 4.390 4.372 4.343 4.302 4.249 4.184 4.106 4.016 3.913 3.797
40 0.217 0.218 0.219 0.220 0.222 0.225 0.228 0.232 0.237 0.243 0.250 40 4.599 4.593 4.575 4.546 4.504 4.450 4.384 4.306 4.215 4.111 3.993
Lateral earth pressure coefficients for vertical frictionless wall with inclined backfill
Wall friction
• Coulomb’s method for walls with inclined back, inclined backfill andfriction.
Ref: B. M. Das (2007)
Active Case
Passive Case
Wall friction
• Simplified solution for Coulomb’s graphical method for lateralearth pressure for walls with inclined back, inclined backfilland friction.
Ref: B. M. Das (2007)
d/f 0.667
f ()
20 0.438 0.474 0.524 0.604 0.907 0.474 0.515 0.570 0.659 0.990 0.514 0.560 0.623 0.721 1.089 0.559 0.612 0.683 0.795 1.208 0.611 0.672 0.754 0.882 1.355
21 0.421 0.455 0.501 0.572 0.736 0.457 0.496 0.548 0.626 0.807 0.498 0.542 0.600 0.689 0.890 0.544 0.594 0.661 0.761 0.989 0.596 0.655 0.732 0.847 1.109
22 0.405 0.437 0.480 0.543 0.670 0.442 0.478 0.526 0.597 0.738 0.483 0.524 0.579 0.659 0.817 0.529 0.577 0.639 0.731 0.911 0.581 0.637 0.710 0.816 1.023
23 0.390 0.420 0.459 0.516 0.621 0.427 0.461 0.506 0.570 0.687 0.468 0.507 0.558 0.631 0.764 0.514 0.560 0.619 0.703 0.854 0.567 0.621 0.690 0.787 0.962
24 0.375 0.403 0.440 0.491 0.581 0.412 0.444 0.486 0.545 0.645 0.453 0.491 0.539 0.606 0.720 0.500 0.544 0.599 0.677 0.807 0.553 0.605 0.670 0.761 0.913
25 0.361 0.387 0.421 0.468 0.546 0.398 0.429 0.467 0.521 0.609 0.440 0.475 0.520 0.582 0.682 0.486 0.528 0.581 0.652 0.768 0.540 0.589 0.651 0.736 0.870
26 0.347 0.372 0.403 0.446 0.514 0.385 0.413 0.450 0.499 0.576 0.426 0.460 0.502 0.559 0.648 0.473 0.513 0.563 0.629 0.732 0.527 0.575 0.633 0.712 0.833
27 0.334 0.357 0.386 0.426 0.486 0.371 0.399 0.433 0.478 0.547 0.413 0.445 0.485 0.538 0.618 0.460 0.498 0.545 0.608 0.701 0.515 0.560 0.616 0.690 0.800
28 0.321 0.343 0.370 0.406 0.460 0.359 0.385 0.416 0.458 0.521 0.401 0.431 0.469 0.518 0.590 0.448 0.484 0.529 0.587 0.671 0.503 0.546 0.599 0.668 0.769
29 0.309 0.330 0.355 0.388 0.436 0.347 0.371 0.401 0.440 0.496 0.389 0.417 0.453 0.499 0.564 0.436 0.471 0.513 0.567 0.645 0.491 0.533 0.583 0.648 0.741
30 0.297 0.316 0.340 0.371 0.414 0.335 0.358 0.386 0.422 0.473 0.377 0.404 0.438 0.480 0.540 0.425 0.458 0.497 0.548 0.620 0.479 0.519 0.568 0.629 0.714
31 0.286 0.304 0.326 0.354 0.394 0.324 0.345 0.371 0.405 0.451 0.366 0.392 0.423 0.463 0.518 0.413 0.445 0.483 0.530 0.596 0.468 0.507 0.553 0.611 0.690
32 0.275 0.292 0.312 0.338 0.374 0.313 0.333 0.357 0.389 0.431 0.355 0.379 0.409 0.446 0.497 0.402 0.432 0.468 0.513 0.574 0.457 0.494 0.538 0.593 0.667
33 0.264 0.280 0.299 0.323 0.356 0.302 0.321 0.344 0.373 0.412 0.344 0.367 0.395 0.430 0.477 0.392 0.420 0.454 0.497 0.553 0.447 0.482 0.524 0.576 0.645
34 0.254 0.269 0.287 0.309 0.339 0.292 0.310 0.331 0.358 0.394 0.334 0.356 0.382 0.415 0.458 0.381 0.409 0.441 0.481 0.534 0.437 0.471 0.511 0.560 0.624
35 0.244 0.258 0.275 0.295 0.323 0.282 0.299 0.319 0.344 0.377 0.323 0.345 0.370 0.400 0.440 0.371 0.397 0.428 0.466 0.515 0.427 0.459 0.498 0.544 0.604
36 0.235 0.248 0.263 0.282 0.307 0.272 0.288 0.307 0.331 0.361 0.314 0.334 0.357 0.386 0.423 0.362 0.387 0.416 0.451 0.497 0.417 0.448 0.485 0.529 0.586
37 0.226 0.238 0.252 0.270 0.293 0.263 0.278 0.296 0.317 0.345 0.304 0.323 0.346 0.373 0.407 0.352 0.376 0.404 0.437 0.480 0.408 0.438 0.473 0.515 0.568
38 0.217 0.228 0.242 0.258 0.279 0.253 0.268 0.285 0.305 0.331 0.295 0.313 0.334 0.360 0.392 0.343 0.366 0.392 0.424 0.464 0.398 0.427 0.461 0.501 0.551
39 0.208 0.219 0.231 0.246 0.265 0.245 0.258 0.274 0.293 0.317 0.286 0.303 0.323 0.347 0.377 0.334 0.356 0.381 0.411 0.448 0.389 0.417 0.449 0.487 0.534
40 0.200 0.210 0.221 0.235 0.253 0.236 0.249 0.264 0.281 0.303 0.277 0.294 0.312 0.335 0.363 0.325 0.346 0.370 0.398 0.433 0.381 0.407 0.438 0.474 0.518
Active earth pressure coefficients for a rough wall with inclined backfill
Passive earth pressure coefficients for a rough wall with inclined backfill
d/f 0.100
f ()
20 2.144 2.444 2.780 3.167 3.626 2.016 2.288 2.586 2.922 3.310 1.921 2.174 2.445 2.746 3.085 1.854 2.093 2.348 2.624 2.930 1.810 2.043 2.287 2.547 2.831
21 2.233 2.554 2.916 3.337 3.842 2.093 2.381 2.701 3.064 3.487 1.987 2.254 2.544 2.866 3.234 1.912 2.164 2.433 2.728 3.058 1.861 2.105 2.363 2.639 2.943
22 2.327 2.670 3.061 3.519 4.074 2.173 2.479 2.822 3.214 3.676 2.057 2.338 2.647 2.993 3.391 1.972 2.237 2.523 2.838 3.192 1.915 2.170 2.442 2.736 3.061
23 2.426 2.793 3.214 3.713 4.324 2.256 2.583 2.950 3.373 3.877 2.129 2.426 2.755 3.127 3.558 2.035 2.314 2.617 2.953 3.334 1.970 2.238 2.524 2.836 3.184
24 2.530 2.922 3.377 3.921 4.593 2.344 2.691 3.085 3.543 4.092 2.204 2.519 2.869 3.269 3.735 2.100 2.394 2.715 3.074 3.484 2.027 2.308 2.610 2.942 3.313
25 2.640 3.060 3.551 4.144 4.885 2.437 2.805 3.227 3.723 4.323 2.282 2.616 2.989 3.419 3.924 2.168 2.477 2.818 3.201 3.642 2.087 2.381 2.699 3.052 3.449
26 2.756 3.206 3.737 4.384 5.200 2.533 2.926 3.379 3.915 4.570 2.365 2.717 3.116 3.577 4.124 2.239 2.564 2.925 3.335 3.809 2.149 2.457 2.793 3.167 3.592
27 2.878 3.360 3.935 4.641 5.542 2.635 3.053 3.539 4.120 4.836 2.451 2.824 3.249 3.745 4.338 2.313 2.656 3.039 3.476 3.986 2.213 2.536 2.891 3.288 3.743
28 3.008 3.525 4.146 4.918 5.915 2.742 3.188 3.710 4.339 5.122 2.541 2.936 3.390 3.923 4.567 2.390 2.751 3.157 3.625 4.174 2.280 2.618 2.993 3.415 3.901
29 3.145 3.699 4.373 5.218 6.321 2.855 3.330 3.891 4.574 5.431 2.635 3.055 3.539 4.113 4.811 2.470 2.851 3.282 3.782 4.373 2.349 2.704 3.100 3.549 4.069
30 3.290 3.886 4.616 5.542 6.764 2.973 3.480 4.084 4.825 5.765 2.734 3.179 3.696 4.314 5.072 2.554 2.956 3.414 3.948 4.586 2.422 2.794 3.212 3.689 4.245
31 3.444 4.085 4.877 5.893 7.250 3.099 3.640 4.290 5.095 6.126 2.838 3.310 3.863 4.529 5.353 2.642 3.066 3.552 4.124 4.811 2.497 2.888 3.329 3.837 4.433
32 3.607 4.297 5.158 6.274 7.784 3.231 3.809 4.510 5.385 6.519 2.947 3.448 4.040 4.758 5.655 2.734 3.181 3.698 4.310 5.052 2.575 2.986 3.453 3.992 4.631
33 3.780 4.524 5.461 6.689 8.373 3.371 3.989 4.745 5.697 6.945 3.062 3.594 4.228 5.003 5.980 2.830 3.302 3.853 4.508 5.309 2.657 3.089 3.582 4.157 4.841
34 3.965 4.768 5.789 7.141 9.023 3.518 4.180 4.996 6.035 7.411 3.183 3.749 4.428 5.265 6.330 2.931 3.430 4.016 4.718 5.584 2.742 3.196 3.718 4.330 5.064
35 4.162 5.029 6.143 7.636 9.745 3.675 4.384 5.266 6.400 7.920 3.310 3.912 4.640 5.546 6.709 3.036 3.564 4.188 4.942 5.879 2.831 3.309 3.861 4.513 5.302
36 4.372 5.309 6.527 8.179 10.548 3.841 4.601 5.556 6.796 8.478 3.444 4.086 4.867 5.847 7.120 3.147 3.706 4.370 5.180 6.195 2.924 3.427 4.012 4.707 5.554
37 4.597 5.612 6.944 8.776 11.445 4.017 4.833 5.868 7.225 9.091 3.586 4.269 5.109 6.171 7.566 3.263 3.855 4.564 5.434 6.535 3.021 3.551 4.171 4.913 5.824
38 4.837 5.938 7.399 9.435 12.451 4.204 5.082 6.204 7.693 9.767 3.735 4.465 5.367 6.521 8.051 3.385 4.013 4.769 5.706 6.901 3.123 3.681 4.339 5.131 6.111
39 5.094 6.290 7.896 10.164 13.586 4.403 5.348 6.567 8.204 10.515 3.893 4.672 5.644 6.898 8.580 3.514 4.179 4.988 5.996 7.296 3.229 3.818 4.516 5.363 6.419
40 5.370 6.671 8.439 10.973 14.870 4.615 5.633 6.960 8.763 11.345 4.061 4.893 5.941 7.306 9.159 3.649 4.355 5.220 6.308 7.724 3.341 3.962 4.703 5.609 6.749