BEARING CAPACITY OF FOOTINGS ON SLOPES

This cue card is based on chapt. 4 sect. 9 of Foundation Analysis and Design (5th ed) by Jospeh E. Bowles. To understand it, the user should have the book. Permission to quote from the book has been granted. Comments or questions concerning the cue card should be directed to the developer, Helmut Schmidhofer, on 0500 818 500 (international replace leading 0 with 61) or email engcomp@pbq.com.au.

The following is a brief summary:

  • The angle of the exit, d-E, with the ground slope, BETA, is taken as 45 - PHI/2.
  • The reduced coefficient NC is based on the ratio of failure lines a-d-e (L0) to a-d-E (L1), ie. NCP=NC*L1/L0
  • The reduced overburden coefficient NQ is based on the ratio of overburden areas A0 to A1, ie. NQP=NQ*A1/A0
  • A1 begins to exceed A0 when the berm width BB is approximately 1.5 to 2 times B, in which case no modification to NQ.
  • No reduction in NG if BB/B ³ 2, otherwise:
  • compute Coulomb's passive pressure coefficients for +BETA (Kmax) and -BETA (Kmin) (see bowles11-4);
  • find R=Kmin/Kmax and NDG = NG/2 + NG/2*(R+BB/2*B*(1-R)) where NG is that of Hansen's bearing capacity equations (see hansen-vesic.doc).

Enter your custom values at lines marked ****, then process all lines. The word "process" means to select (highlight) the relevant text and click the spinning globe in the Engineers' Compendium window. To work with the cue card, you need MATHSERV, which you may download here.

STEP 1: The given values are taken from example 4-11:

**** slope in degrees (1 on 2), BETAD = 26.5
**** internal friction in degrees, PHID = 36
**** cohesion in kPa, C = 0
**** unit weight of soil in kN/m3, GAMMA = 14.85

STEP 2: Basic coefficients by Hansen's method. Process the next six lines:

45 degrees in radians, A45 = 45*_pi/180 := 0.7854
internal friction in radians, PHI = PHID*_pi/180 := 0.6283
passive pressure coefficient, KP = (tan(A45+PHI/2))^2 := 3.8518
Coefficient NQ = round(exp(_pi*tan(PHI))*KP) := 38
Coefficient NG = round(1.5*(NQ-1)*tan(PHI)) := 40
Coefficient NC = round((NQ-1)/tan(PHI)) := 51

STEP 3: Modification of NG. Process the next 13 lines:

vertical face in radians, ALPHA = 90*_pi/180 := 1.5708
ground slope in radians, BETA = BETAD*_pi/180 := 0.4625
friction angle DELTA = PHI := 0.6283
temporary TMP1 = sin(ALPHA-PHI)^2 := 0.6545
TMP2 = sin(ALPHA)^2*sin(ALPHA+DELTA) := 0.809
TMP3 = sin(PHI+DELTA)*sin(PHI+BETA)/(sin(ALPHA+DELTA)*sin(ALPHA+BETA)) := 1.1652
TMP4 = sin(PHI+DELTA)*sin(PHI-BETA)/(sin(ALPHA+DELTA)*sin(ALPHA-BETA)) := 0.2168
KP at +BETA, KMAX = TMP1/(TMP2*(1-sqrt(TMP3))^2) := 128.2399
KP at -BETA, KMIN = TMP1/(TMP2*(1-sqrt(TMP4))^2) := 2.8331
ratio R = KMIN/KMAX := 0.02209
two cases of berm ratio BB/B, BR = {.75, 1.5}
modified NDG = NG/2 + NG/2*(R + BR/2*(1-R))
rowprint using "#####"; NDG
   28   35

STEP 4: Modification of NQ and NC. The book refers to Table 4.7 (to be interpolated) or computer program B-2, which is called SMBRGNP.EXE. Two cases for D/B are given and the answer is sought in terms of B. Enter values from Table 4.7 that correspond with your data, then process all lines:

**** for D/B 1.5, BB/B .75, BETA 25, PHI 30, NQ2530 = 16.2
**** ditto, PHI 40, NQ2540 = 42.35
**** ditto, BETA 30, PHI 30, NQ3030 = 15.85
**** ditto, PHI 40, NQ3040 = 40.23
interpolate column, NQ2536 = NQ2530 + (NQ2540-NQ2530)*((PHID-30)/10)^1.5 := 28.3534
interpolate column, NQ3036 = NQ3030 + (NQ3040-NQ3030)*((PHID-30)/10)^1.5 := 27.1808
interpolate row, NQP = round(NQ2536 + (NQ3036-NQ2536)*(BETAD-25)/5) := 28
the computer program returns 27.81, the book uses NQP = 27

STEP 5: Repeat STEP 4 for NC. This is not required for the given example because C = 0. Process the next 3 lines:

**** NCP not required, NCP = 0
depth D is 1.5B, QBAR in terms of B, QBAR = 1.5*GAMMA := 22.275
ultimate in terms of B, QULT = C*NCP + QBAR*NQP + .5*GAMMA*NDG[1] := 807.6629

NOTE: The result QBAR = 808B corresponds with the test result of 891B. Shape factors SQ and SG are 1.0 for strip footing. If pad footing, they should be computed (see hansen-vesic.doc).