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BFC 4043
2.0 SHALLOW FOUNDATION :
2.1 General Concept
A shallow foundation must :
- be safe against overall shear failure in the soil- not undergo excessive settlement
Nature of bearing capacity failure are : (as shown in Figure 2.1)
- general shear failure (for stiff clay or dense sand)- local shear failure (for medium dense sand or
clayey soil)- punching shear failure(loose sand or soft clay)
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Figure 2.1 Nature of bearing capacity failure : (a) general shear (b) local shear (c) punching shear
Vesic (1973) proposed a relationship for the bearing capacity failure on sands in terms of relative density, Dr
depth of foundation, Df and B*, Figure 2.2
Where : and B – width, L – length of
foundation
NOTE : L IS ALWAYS GREATER THAN B
For square; B=L and for circular; B=L=Diameter of foundation and B* = B
Figure 2.2 Modes of foundation failure in sand, (Vesic, 1973)
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2.2 Terzaghi’s Bearing Capacity
Terzaghi suggested for a continuous or strip foundation with failure surface as in Figure 2.3
Figure 2.3 Bearing capacity failure in soil under rough rigid continuous foundation
Soil above the bottom of foundation is surcharge, q = Df
The failure zone under the foundation is separated into three parts namely;
- triangular ACD under the foundation- radial shear zones ADF and CDE with curves DE
and DF as arcs of logarithmic spiral- Rankine passive zones AFH and CEG
CAD and ACD are assume to equal friction angle, Ø
Thus ultimate bearing capacity, qu for general shear failure can be expressed as :
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Where : c – cohesion of soil - unit weight of soil
q = Df
Nc, Nq, N - bearing capacity factors
And
where - passive pressure coefficient
Table 2.1 summarizes values for Nc, Nq, and N
Table 2.1 Terzaghi’s Bearing Capacity’s Factors
Ø Nc Nq N Ø Nc Nq N0123456789
10111213141516171819202122232425
5.706.006.306.626.977.347.738.158.609.099.61
10.1610.7611.4112.1112.8613.6814.6015.1216.5617.6918.9220.2721.7523.3625.13
1.001.101.221.351.491.641.812.002.212.442.692.983.293.634.024.454.925.456.046.707.448.269.19
10.2311.4012.72
0.000.010.040.060.100.140.200.270.350.440.560.690.851.041.261.521.822.182.593.073.644.315.096.007.088.34
26272829303132333435363738394041424344454647484950
27.0929.2431.6134.2437.1640.4144.0448.0952.6457.7563.5370.0177.5085.9795.66
106.81119.67134.58151.95172.28196.22224.55258.28298.71347.50
14.2115.9017.8119.9822.4625.2828.5232.2336.5041.4447.1653.8061.5570.6181.2793.85
108.75126.50147.74173.28204.19241.80287.85344.63415.14
9.8411.6013.7016.1819.1322.6526.8731.9438.0445.4154.3665.2778.6195.03
115.31140.51171.99211.56261.60325.34407.11512.84650.67831.99
1072.80
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From Kumbhojkar (1993)
And ultimate bearing capacity, qu for local shear failure can be expressed as :
Where : N’c, N’q, N’ (see Table 2.2) are reduced bearing capacity factors can be calculated by using N’c, N’q, N’ - bearing capacity factors with
Table 2.2 Terzaghi’s Modified Bearing Capacity’s Factors
Ø N’c N’q N’ Ø N’c N’q N’0123456789
10111213141516171819202122232425
5.705.906.106.306.516.746.977.227.477.748.028.328.638.969.319.67
10.0610.4710.9011.3611.8512.3712.9213.5114.1414.80
1.001.071.141.221.301.391.491.591.701.821.942.082.222.382.552.732.923.133.363.613.884.174.484.825.205.60
0.000.0050.020.04
0.0550.0740.10
0.1280.160.200.240.300.350.420.480.570.670.760.881.031.121.351.551.741.972.25
26272829303132333435363738394041424344454647484950
15.5316.3017.1318.0318.9920.0321.1622.3923.7225.1826.7728.5130.4332.5334.8737.4540.3343.5447.1351.1755.7360.9166.8073.5581.31
6.056.547.077.668.319.039.82
10.6911.6712.7513.9715.3216.8518.5620.5022.7025.2128.0631.3435.1139.4844.4550.4657.4165.60
2.592.883.293.764.394.835.516.327.228.359.41
10.9012.7514.7117.2219.7522.5026.2530.4036.0041.7049.3059.2571.4585.75
Example 2.1
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Given : A square foundation, 1.5m x 1.5m in plan view
Soil parameters : Ø’ = 20°, c’ = 15.2 kN/m2, =17.8 kN/m3
Assume : FS = 4, general shear failure condition and Df= 1 mFind : Allowable gross load on the foundation
Solution :
For Ø’ = 20°, (Table 2.1); Nc = 17.69, Nq = 7.44, N = 3.64
Thus
Allowable bearing capacity :
Thus total allowable gross load, Q
Example 2.2
Given : Repeat example 2.1Assume : Local shear failure condition Solution :
For Ø’ = 20°, (Table 2.2); Nc = 11.85, Nq = 3.88, N = 1.12
Allowable load :
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;
2.3 Effect of Water Table on Bearing Capacity
All equations mentioned before are based on the location of water table well below the foundation; if otherwise, some modification should be made according to the location of the water table, see Figure 2.4
Figure 2.4 Modification of bearing capacity for water table
Case I : 0 ≤ D1 ≤ Df
- q’(effective surcharge) = - where :- - effective unit weight = - - saturated unit weight of soil- - unit weight of water = 9.81kN/m3 or 62.4 lb/ft3
- in the last term of the equation Case II : 0 ≤ d ≤ B
- the value
-
Case III : d ≥ B- water has no effect on the qu
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Note : the values of bearing capacity factors used strictly depending on whether the condition is general or local shear failure.
2.4 Factor of Safety, FS
, where :
- qall - gross allowable load-bearing capacity,- qu – gross ultimate bearing capacity,- FS – factor of safety
Values of FS against bearing capacity failure is 2.5 to 3.0.
Net stress increase on soil = net ultimate bearing capacity/FS
, and : ;
Where : qall(net) – net allowable bearing capacity qu(net) – net ultimate bearing capacity
Procedure for FSshear
a. Find developed cohesion,cd and angle of friction,Ød;
b. Terzaghi’s equations become (with cd and Ød):
With : Nc, Nq, N - bearing capacity factors for Ød
c. Thus, the net allowable bearing capacity :
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Example 2.3
Using ; and FS = 5; find net allowable load for the
foundation in example 2.1 with qu = 521 kN/m2
With qu = 521 kN/m2; q = 1(17.8) = 17.8 kN/m2
Hence Qall(net) = 100.64(1.5x1.5) = 226.4 kN
Example 2.4
Using Example 3.1, and Terzaghi’s equation
with FSshear = 1.5;
Find net allowable load for the foundation
For c=15.2 kN/m2, Ø = 20° and
cd =
Ød = tan-1[ ] = tan-1[ ] = 13.64°
With :
From Table 2.1 : Ø=13.6° ; ; ; (estimation)
Hence :
2.5 The General Bearing Capacity Equation
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The need to address for rectangular shape foundation where :(0<B/L<1)
Meyerhof (1963) suggest :
Where :
c – cohesionq – effective stress at the level of the bottom of
foundation - unit weight of soil
B – width (or diameter) of foundationNc, Nq, N - bearing capacity factors
- shape factors- depth factors
- load inclination factors
Values
- bearing capacity factors :
- shape, depth and inclination factors :- shape
Where : L – length of the foundation and (L>B)
- depthif Df/B ≤ 1
if Df/B > 1
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NOTE : tan-1(Df/B) is in radian
- inclination
Where : β – inclination of load from vertical
For undrained condition (Ø = 0)
Skempton’s :
Table 2.3 Vesic’s Bearing Capacity Factors for General Equation (1973)
Ø Nc Nq N Nq/ Nc Tan Ø Ø Nc Nq N Nq/ Nc Tan Ø
0123456789
10111213141516171819202122232425
5.145.385.635.906.196.496.817.167.537.928.358.809.289.81
10.3710.9811.6312.3413.1013.9314.8315.8216.8818.0519.3220.72
1.001.091.201.311.431.571.721.882.062.252.472.712.973.263.593.944.344.775.265.806.407.077.828.669.60
10.66
0.000.070.150.240.340.450.570.710.861.031.221.441.691.972.292.653.063.534.074.685.396.207.138.209.44
10.88
0.200.200.210.220.230.240.250.260.270.280.300.310.320.330.350.360.370.390.400.420.430.450.460.480.500.51
0.000.020.030.050.070.090.110.120.140.160.180.190.210.230.250.270.290.310.320.340.360.380.400.420.450.47
26272829303132333435363738394041424344454647484950
22.2523.9425.8027.8630.1432.6735.4938.6442.1646.1250.5955.6361.3567.8775.3183.8693.71105.1
1118.3
7133.8
8152.1
0173.6
4199.2
6229.9
3266.8
9
11.8513.2014.7216.4418.4020.6723.1826.0929.4433.3037.7542.9248.9355.9664.2073.9085.3899.02115.3
1134.8
8158.5
1187.2
1222.3
1265.5
1319.0
7
12.5414.4716.7219.3422.4025.9930.2235.1941.0648.0356.3166.1978.0392.25109.4
1130.2
2155.5
5186.5
4224.6
4271.7
6330.3
5403.6
7496.0
1613.1
0.530.550.570.590.610.630.650.680.700.720.750.770.800.820.850.880.910.940.971.011.041.081.121.151.20
0.490.510.530.550.580.600.620.650.670.700.730.750.780.810.840.870.900.930.971.001.041.071.111.151.19
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6762.8
9
Example 2.5
Figure 2.5
Given : A square foundation (B x B), Figure 2.5, Q=150 kN. Df = 0.7m, load is inclined at 20˚ from vertical, FS = 3. Use general bearing capacity factors
Find : The width of foundation B
;
From Table 2.3 : For Ø’ =30°: Nq = 18.4, N = 22.4, Nq/ Nc = 0.61, Tan Ø = 0.58
;
;
;
So
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By trial and error : B=1.3m
2.6 Eccentrically Loaded Foundations
Eccentrically loaded foundations give non-uniform distribution of pressure, Figure 2.6
Figure 2.6 Eccentrically loaded foundations
Eccentricity,
qmax and qmin is given by :
and
if e > B/6, and qmin becomes negative then :
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Factor of safety against bearing capacity failure; effective area method, by Meyerhof (1953)a. Find effective dimensions of the dimensions
- the smaller of B’ and L’ is the width- effective width, B’ = B – 2e- effective length, L’ = L- if e is in the direction of L than L’ = L – 2e
b. Find the ultimate bearing capacity, qu :
- use L’ and B’ to find - use B to find
c. Total ultimate load, ; where A’ – effective area
d. Factor of safety,
e. Check FS against qmax ;
Example 2.6
Given : A square foundation as shown in Figure 2.7. Using general bearing capacity factors, (table 2.3)
Figure 2.7Find : Ultimate load, Qult, assume one way load eccentricity, e = 0.15m
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Sand :
1.5m x 1.5 m
0.7 m
BFC 4043
Solution : with c = 0;
Where :
q = 0.7(18) = 12.6 kN/m2
for Ø = 30°, from Table 2.3 : Nq=18.4 and N =22.4B’ = 1.5 – 2(0.15) = 1.2mL’ = 1.5m
Thus values for general bering capacity equations : (using B’ and L’)
Qult = q’u X A’ = 549.2 X (1.5X1.2) = 988kN
Qall = 988/3 = 330kN with FS=3
2.7 Load on strip footing
Example 2.7 :
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Given : The strip footing shown below is to be constructed in a uniform deposit of stiff clay and must support a wall that imposes a loading of 152 kN/m of wall length. Use general bearing capacity factors.
Find : The width of footing with FS of 3.
Figure 2.8Solution :
And Ø=0°; from the Table 2.3 Nc = 5.14, Nq = 1.0 and N =0
B required is 1.5 meter to be conservative
2.8 Dimension of loaded square pad footing
Example 2.8 :Soil deposit has the following ; =20.44 kN/m3, Ø=30°, c=38.3kN/m2
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Square footing located 1.52 m below surface, carries 2670 kN and groundwater is negligible. Use Terzaghi’s values, (Table 2.1).
Find : The right dimension B. Use Terzaghi’s equation
With Ø=30°; Nc=37.16, Nq=22.46, N =19.13
Assume B=3 m;
Assume B=1 m;
Assume B=2m;
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2670 kN
γ = 20.44 kN/m3
1.52m Ø=30˚c = 38.3 kN/m2
Figure 2.9
BFC 4043
Assume B=1.8m;
Assume B=1.7m;
Therefore use 1.7m x 1.7m
2.9 Contact Pressure and stability check.
Can be computed by using flexural formula of :
Where :
q – contact pressureQ – total axial vertical loadA – area of footing
Mx, My – total moment about respective x and y axes Ix, Iy – moment of inertia about respective x and y axes
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x, y – distance from centroid to the outer most point at which the contact pressure is computed along respective x and y axes.
Example 2.9
A pad footing with dimension of 1.52 x 1.52m acted upon by the load of 222.4kN. Estimate soil contact pressure and FS against bearing capacity.
Given :1.52m by 1.52m square footing; P=222.4kN;
=18.85kN/m3
=24 kN/m3; qu = 143.64 kN/m2
Find :a. Soil contact pressure b. FS against bearing capacity pressure
Solution :
a. ; Mx=My=0; since load on centroid
Total load calculation, Q :
Column load, P = 222.4kNWeight of footing base = (1.52m)(1.52m)0.31m(24kN/m3) = 17.19 kNWeight of footing pedestal = (0.14m)(0.14m)(0.91m)(24kN/m3) = 0.43 kNWeight of backfill soil
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222.4KN
0.14m20.91m
1.22m
0.31m1.52m
Figure 2.10
BFC 4043
= [(1.52m)(1.52m)-(0.14m)(0.14m)](0.91m) x 18.85kN/m3
= 39.3kN
Q = 222.4 + 17.19 + 0.43 + 39.3 = 279.32kNArea, A = 1.52mx1.52m = 2.31m2
Soil contact pressure or Stress, q = Q/A = 120.9 kN/m 2
b.
Assuming cohesive soil has : Ø=0° and c>0; thus :Nc=5.14, Nq=1.0, N =0, Df=1.22m
Since FS > 3.0; thus ok.
Given : 2.29m by 1.52m rectangular footing
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Example 2.10
Draw soil contact pressure for footing in Figure 2.11
Conversion to SI unitP=222.4 kN; H=88.96 kN; M=81.35kN.m; W=88.96 kNDf=1.22m; B=2.29m (7.5ft);L=1.52m (5ft)
Figure 2.11
BFC 4043
Find : Contact pressure and soil pressure diagram
Solution :
Using flexural formula;
Q = P + W = 222.4 kN + 88.96 kN = 311.36 kN.A = 2.29m x 1.52m = 3.48 m2; Mx=0; My=88.96(1.22)+81.35=189.88kN.m (Moment about point C)
x = 2.29/2 = 1.145m;
Take ΣV = 0 and ΣMc = 0 will produce :
ΣV = 0 :
ΣMc = 0 : see Figure 2.12 (b) and (c)
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Example 2.11 Checking stability on shallow foundation
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Figure 2.12 (a) and (b)
1.61m
254.46kN/m2
2.29m2.29m
BFC 4043
Given : A 6 ft x 6 ft footing as shown; load P=60kips; weight of concrete footing including pedestal + base pad, W1=9.3kips; backfill, W2=11.2kips; horizontal load = 4kips; qall for soil = 3.0 kips/ft2.
Find :1. Contact pressure and soil pressure diagram.2. Shear and moment at section A-A (in the Figure E3.14)3. FS against sliding if coefficient of friction, δ = 0.404. FS against overturning.
Solution :
1.
Q=P+W1+W2=60+9.3+11.2=80.5kipsA=6ftx6ft=36ft2
My=4kipsx4.5ft=18kip-ft (about point C)x=6ft/2=3ftIy=6ft(6ft)3/12=108ft4; Mx=0; Mxy/Ix=0
So : qright = 2.74 kips/ft2 < 3.0 kips/ft2 ; OK qleft = 1.74 kips/ft2 < 3.0 kips/ft2 ; OK
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Figure 2.13
BFC 4043
2. ΔFDG and ΔEDH are similar triangles; so
3.
4.
2.10Settlement of shallow foundation
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Pressure diagram
BFC 4043
Foundation settlement under load can be classified according to two major types :
(a) immediate or elastic settlement, Se
(b) consolidation settlement, Sc
Elastic settlement, Se takes place immediately during or after construction of structure.
Consolidation settlement, Sc is time dependent comprises of two phases; namely, primary and secondary consolidation settlement.
2.10.1 Elastic settlement of foundations on saturated clay
Elastic settlement of foundations on saturated clay is given by Janbu et al., (1956) using the equation :
where :
A1 is a function of H/B and L/B and A2 is a function of Df/B
All parameters of H, B and Df (with L into the paper) are as
shown in Figure 2.14.
Figure 2.14 : Parameters
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Figure 2.15 : A2 Versus Df/B
Figure 2.16 : A1 Versus H/B and L/B
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Elastic settlement of foundation on sand based on SPT N- valueBy MeyerhofRefer section 5.13 page 263 Equations 5.61 and 5.62
2.10.2 Elastic settlement of foundations on sandy soil: use of strain influence factor
Schmertmann, (1978) proposed that the elastic settlement in sandy soil as :
where :
Iz – strain influence factor
C1 – correction factor due to depth =
C2 – correction factor due to soil creep =
- stress at the level of foundation (due to loading + self weight of footing + weight of soil above footing)
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Figure 2.17 : Calculation of elastic settlement using strain influence factor
The variation of Iz with depth below the footing for square or circular are as below :
Iz = 0.1 at z = 0Iz = 0.5 at z = z1 = 0.5BIz = 0 at z = z2 = 2B
Footing with L/B ≥ 10 (rectangular footing) :
Iz = 0.2 at z = 0Iz = 0.5 at z = z1 = BIz = 0 at z = z2 = 4B
2.10.3 Range of material parameters
Elastic parameters such as Es and μs in Table 2.4 can be used if the real laboratory test results not available.
Table 2.4 : Elastic parameters of various soils
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Type of soil Modulus of Elasticity, Es
(MN/m2)
Poisson’s ratio, μs
Loose sand 10.5 – 24.0 0.20 – 0.40Medium dense
sand17.25 – 27.60 0.25 – 0.40
Dense sand 34.50 – 55.20 0.30 – 0.45Silty sand 10.35 – 17.25 0.20 – 0.40
Sand and gravel 69.00 – 172.50 0.15 – 0.35Soft clay 4.1 – 20.7
0.20 – 0.50 Medium clay 20.7 – 41.4Stiff clay 41.4 – 96.6
2.10.4 Consolidation settlement
(a) Primary consolidation, Sc
Many methods were developed in estimating the value of consolidation settlement, Sc.
Due to simplicity only chart based on Newmarks (1942), Figure 2.18 will be used in estimating the consolidation settlement.
Primary consolidation, Sc calculated as :
where : Cc – compression index (given)H – thickness of clay layere0 – initial void ratio (given)p = p0 + Δp, final pressure
p0 – overburden pressureΔp =4(Ip)q0 – net consolidation pressure at mid-
height of clay layer
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Ip – Influence factor (from Figure 2.18)q0 – net stress increase
Figure 2.18 : Chart for determining stresses below corners of rigid foundation and isotropic soil
Example 2.7
Given :
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Figure 2.19
A foundation to be constructed as in Figure 2.19. The base of the foundation is 3m by 6m, and it exerts a total load of 5400 kN, which include all self weight. The initial void ratio, e0 is 1.38 and compression index, Cc is 0.68.
Required :
Expected primary consolidation settlement of clay layer.
Solution :
p0 = 19.83(200 - 198) + (19.83 – 9.81)(198 - 192) + (17.1 – 9.81)(192 – 185.6)/2 = 123.1 kN/m2
Weight of excavation = 19.83(200 - 198) + (19.83 – 9.81)(198 – 195.5) = 64.7kN/m2
By dividing the base into 4 equal size of 1.5m by 3.0m :
mz = 1.5m nz = 3.0m
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;
From Figure 2.18, the influence coefficient is 0.04
Therefore ;
Final pressure, p = p0 + Δp = 123.1 + 37.6 = 160.7 kN/m2.
Therefore;
(b) secondary consolidation
Secondary settlement, Ss is computed from the following calculation (U.S. Department of the Navy, 1971)
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where :
Ss – secondary compression settlementCα – coefficient of secondary compression, can be
determined from Figure 3.26H – thickness of clay layer that is considered ts – time for which settlement is requiredtp – time to completion of primary consolidation
Figure 2.20 : Value of Cα
2.11 Allowable bearing pressure in sand based on settlement consideration.
Bowles (1977) proposed a correlation of the net allowable bearing pressure for foundations with SPT (N-values).
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The following equations are used :
And
Where :
Fd – depth factor =
S – tolerable settlement, in mm.
Example 2.8
Given:
A shallow square footing for a column is to be constructed. Design load is 1000 kN. The foundation soil is sand. The SPT numbers from field exploration as shown in the table.
Assume that the footing must be 1.5m deep, the tolerable settlement as 25.4mm and the size is > 1.22m.
Required :
(a) The exact size of the footing (b) safety factor for foundation
Solution :
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Navg = (7+8+11+11+13+10+9+10+12)/9=10With S=25.4mm and N=10
By trial and error (set the table for calculation)
From the table it is seen that the appropriate B=2.4
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Setting general equation and equation for net ultimate with c=0 (for sandy soil) :
;
For N=10; friction angle of Ø=34˚ is considered (from table on SI)
With no inclination so Fqi=Fγi=1.0From table 2.3 Nq=29.44, Nγ=41.06
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So for a tolerable settlement of 25.4mm, the SF required is calculated as : SF=Qnet(u)/Q= 10,322kN/1000kN = 10.3 which is OK, therefore most design controlled by tolerable criterion.
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