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Lectures 26 - 31 _Remaining Lectures_

# Lectures 26 - 31 _Remaining Lectures_ - EP11 Rankine...

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EP11 P a or P p i H/3 i Rankine Inclined Backfill Surface – Cohesionless Soil (sands) o Plane parallel to soil surface has both a shear stress and a normal stress o Plane parallel to wall surface has both a shear stress and a normal stress o Use same Mohr circle approach - look at a small cube of soil, width b at depth z, with an inclined surface: W = γ bz & A = b/cos i σ v = W/A = γ zcos i o Note in Mohr circle plot that i max = φ Can the wall provide this much friction? o We still get a triangular pressure distribution but at an incline i. o The resultant force will be inclined at angle i also but still located at H/3 above base. o p = K σ v o P = ½K γ H 2 o φ - + φ - - = cos i cos i cos cos i cos i cos i cos K 2 2 2 2 a o φ - - φ - + = cos i cos i cos cos i cos i cos i cos K 2 2 2 2 p γ , φ , c A H B i z σ v = γ z cos i σ a or σ p b Shear Stress, τ Normal Stress, σ ( σ i , τ i ) φ τ f = σ tan φ Pole Failure Plane ( σ a , τ a ) φ Failure Plane i Active Case ( σ i , τ i ) φ Pole Failure Plane ( σ p , τ p ) φ Failure Plane i Passive Case

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EP12 Rankine Inclined Backfill Example Find the Active Earth Pressure: 282 . 0 35 cos 10 cos 10 cos 35 cos 10 cos 10 cos 10 cos K 2 2 2 2 a = ° - ° + ° ° - ° - ° ° = p a = K a γ H = (0.282)(120 pcf)(20’) = 677 psf P a = ½K a γ H 2 = ½ (0.282)(120 pcf)(20’) 2 = 6,768 plf P h = P a cos i = 6,768 cos 10° = 6,665 plf P v = P a sin i = 6,768 sin 10° = 1,175 plf Sand γ = 120 pcf φ = 35° c = 0 A 20’ B 10° z 20’ P a = 6,768 plf 20’/3 = 6.67 0 677 psf i = 10°
EP13 Rankine Earth Pressure Coefficients φ Active Pressure Coefficient, K a , for Backfill Slope Angle, i (degrees) degrees 10° 15° 20° 25° 30° 26° 0.3905 0.3959 0.4134 0.4480 0.5152 0.6999 28° 0.3610 0.3656 0.3802 0.4086 0.4605 0.5727 30° 0.3333 0.3372 0.3495 0.3729 0.4142 0.4936 0.8660 32° 0.3073 0.3105 0.3210 0.3405 0.3739 0.4336 0.5741 34° 0.2827 0.2855 0.2944 0.3108 0.3381 0.3847 0.4776 36° 0.2596 0.2620 0.2696 0.2834 0.3060 0.3431 0.4105 38° 0.2379 0.2399 0.2464 0.2581 0.2769 0.3070 0.3582 40° 0.2174 0.2192 0.2247 0.2346 0.2504 0.2750 0.3151 42° 0.1982 0.1997 0.2044 0.2129 0.2262 0.2465 0.2784 44° 0.1802 0.1815 0.1855 0.1927 0.2039 0.2207 0.2465 φ Passive Pressure Coefficient, K p , for Backfill Slope Angle, i (degrees) degrees 10° 15° 20° 25° 30° 26° 2.561 2.507 2.346 2.083 1.714 1.174 28° 2.770 2.715 2.551 2.284 1.918 1.434 30° 3.000 2.943 2.775 2.502 2.132 1.664 0.866 32° 3.255 3.196 3.022 2.740 2.362 1.894 1.306 34° 3.537 3.476 3.295 3.002 2.612 2.135 1.570 36° 3.852 3.788 3.598 3.293 2.886 2.394 1.827 38° 4.204 4.136 3.936 3.615 3.189 2.676 2.094 40° 4.599 4.527 4.316 3.977 3.526 2.987 2.380 42° 5.045 4.968 4.744 4.383 3.904 3.333 2.694 44° 5.550 5.468 5.228 4.842 4.331 3.721 3.042

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EP14 External Loads at the Top of a Retaining Wall o Any load on the backfill adds to the effective stress behind the wall. Previous methods provided equations used to estimate the increase in both vertical and horizontal stresses beneath a loaded area on a semi-infinite soil mass and used the Poisson ratio to calculate horizontal stress. Behind a retaining wall the soil mass is not semi-infinite and may deflect,. So, although the form of these equations may be reasonable, they must be modified. Uniform Vertical Surcharge o A uniform vertical surcharge, q v , increases the vertical stress at all points in the soil behind the wall.
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