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NPTEL – ADVANCED FOUNDATION ENGINEERING-I

Module 3

Lecture 9 to 12

SHALLOW FOUNDATIONS: ULTIMATE BEARING CAPACITY

Topics 9.1 INTRODUCTION

9.2 GENERAL CONCEPT

9.3 TERZAGHI’S BEARING CAPACITY THEORY

9.4 MODIFICATION OF BEARING CAPACITY EQUATIONS

FOR WATER TABLE

9.5 MODIFICATION OF BEARING CAPACITY EQUATIONS

FOR WATER TABLE

9.6 CASE HISTORY: ULTIMATE BEARING CAPACITY IN

SATURATED CLAY

Analysis of the Field Test Results

9.7 FACTOR SAFETY

10.1 THE GENERAL BEARING CAPACITY EQUATION

Bearing Capacity Factors General Comments

10.2 EFFECT OF SOIL COMPRESSIBILITY

10.3 ECCENTRICALLY LOADED FOUNDATIONS


Foundation with Two-Way Eccentricity

BEARING CAPACITY OF LAYERED SOILS-STRONGER SOIL UNDERLAIN BY WEAKER SOIL

Special Cases

11.1 BEARING CAPACITY OF FOUNDATIONS ON TOP OF A SLOPE

11.2 SEISMIC BEARING CAPACITY AND SETTLEMENT IN GRANULAR SOIL

12.1RECENT ADVANCES IN BEARING CAPACITY OF FOUNDATIONS ON REINFORCED SOIL

12.2 FOUNDATIONS ON SAND WITH GEOTEXTILE

REINFORCEMENT

12.3 FOUNDATIONS ON SATURATED CLAY (𝝓𝝓 = 𝟎𝟎) WITH

GEOTEXTILE REINFORCEMENT

12.4 FOUNDATIONS ON SAND WITH GEOGRID

REINFORCEMENT

12.5 STRIP FOUNDATIONS ON SATURATED CLAY (𝝓𝝓 =

𝟎𝟎) WITH GEOGRID REINFORCEMENT

PROBLEMS REFERENCES


Module 3

Lecture 9

SHALLOW FOUNDATIONS: ULTIMATE BEARING CAPACITY

Topics

1.1 INTRODUCTION 1.2 GENERAL CONCEPT 1.3 TERZAGHI’S BEARING CAPACITY THEORY 1.4 MODIFICATION OF BEARING CAPACITY EQUATIONS

FOR WATER TABLE 1.5 MODIFICATION OF BEARING CAPACITY EQUATIONS

FOR WATER TABLE 1.6 CASE HISTORY: ULTIMATE BEARING CAPACITY IN

SATURATED CLAY Analysis of the Field Test Results

1.7 FACTOR SAFETY


INTRODUCTION

To perform satisfactorily, shallow foundations must have two main characteristics:

1. The foundation has to be safe against overall shear failure in the soil that supports it.

2. The foundation cannot undergo excessive displacement, that is, settlement. (The term excessive is relative, because the degree of settlement allowable for a structure depends on several considerations).

The load per unit area of the foundation at which the shear failure in soil occurs is called the ultimate bearing capacity, which is the subject of this chapter.

GENERAL CONCEPT

Consider a strip foundation resting on the surface of a dense sand or stiff cohesive soil, as shown in figure 3.1a, with a width of B. Now, if load is gradually applied to the foundation, settlement will increase. The variation of the load per unit area on the foundation, qc , with the foundation settlement is also shown in figure 3.1a. At a certain point-when the load per unit area equals qu − a sudden failure in the soil supporting the foundation will take place, and the failure surface in the soil will extend to the ground surface. This load per unit area, qu , is usually referred to as the ultimate bearing capacity of the foundation. When this type of sudden failure in soil takes place, it is called the general shear failure.

Figure 3.1 Nature of bearing capacity failure in soils: (a) general shear failure; (b) local shear failure; (c) punching shear failure (redrawn after Vesic, 1973)


If the foundation under consideration rests on sand or clayey soil of medium compaction (figure 3.1b), an increase of load on the foundation will also be accompanied by an increase of settlement. However, in this case the failure surface in the soil will gradually extend outward from the foundation, as shown by the solid lines in figure 3.1b. When the load per unit area on the foundation equals qu(1), the foundation movement will be accompanied by sudden jerks. A considerable movement of the foundation is then required for the failure surface in soil to extend to the ground surface (as shown by the broken lines in figure 3.1b). The load per unit area at which this happens is the ultimate bearing capacity, qu . Beyond this point, an increase of load will be accompanied by a large increase of foundation settlement. The load per unit area of the foundation, qu(1) is referred to as the first failure load (Vesic, 1963). Note that a peak value of q is not realized in this type of failure,, which is called the local shear failure in soil.

If the foundation is supported by a fairly loose soil, the load-settlement plot will be like the one in figure 3.1c. In this case, the failure surface in soil will not extend to the ground surface. Beyond the ultimate failure load, qu , the load-settlement plot will be steep and practically linear. This type of failure in soil is called the punching shear failure.

Vesic (1963) conducted several laboratory load-bearing tests on circular and rectangular plates supported by a sand at various relative densities of compaction, Dr ,. The variation of qu(1)/1

2γB and qu /12γB obtained from those tests are shown in figure 3.2 (B =

diameter of circular plate or width of rectangular plate, and γ = dry unit weight of sand). It is important to note from this figure that, for Dr ≥ about 70%, the general shear type of failure in soil occurs.


Figure 3.2 Variation of qu(1)/0.5γB and qu /0.5γB for circular and rectangular plates on the surface of a sand (after Vesic, 1963)

Based on experimental results, Vesic (1973) proposed a relationship for the mode of bearing capacity failure of foundations resting on sands. Figure 3.2 shows this relationship, which involves the notation


Figure 3.3 Modes of foundation failure in sand (after Vesic, 1973)

Dr = relative density of sand

Df = depth of foundation measured from the ground surface

B∗ = 2BLB+L

[3.1]

Where

B = width of foundation

L = length of foundation

(Note: L is always greater than B).

For square foundations, B = L; for circular foundation, B = L = diameter, so

B∗ = B [3.2]

Figure 3.4 shows the settlement, S, of the circular and rectangular plates on the surface of sand at ultimate load as described in figure 3.2. It shows a general range of S/B with the relative density of compaction of sand. So, in general, we can say that for foundations at a shallow depth (that is, small Df/B∗), the ultimate load may occur at a foundation settlement of 4-10% of B. this condition occurs when general shear failure in sol occurs; however, in the case of local or punching shear failure, the ultimate load may occur at settlement of 15-25% of the width of the foundation (B).


Figure 3.4 Range of settlement of circular and rectangular plates at ultimate load (Df/B = 0) in sand (after Vesic, 1963)

TERZAGHI’S BEARING CAPACITY THEORY

Terzaghi (1943) was the first to present a comprehensive theory for the evaluation of the ultimate bearing capacity of rough shallow foundations. According to this theory, a foundation is shallow if the depth, Df (figure 3.5), of the foundation is less than or equal to the width of the foundation. Later investigators, however, have suggested that foundation with Df equal to 3.4 times the width of the foundation may be defined as shallow foundations.


Figure 3.5 Bearing capacity failure in soil under a rough rigid continuous foundation

Terzaghi suggested that for a continuous, or strip foundation (that is, the width-to-length ratio of the foundation approaches zero), the failure surface in soil at ultimate load may be assumed to be similar to that shown in figure 3.5. (Note that this is the case of general shear failure as defined in figure 3.1a). The effect of soil above the bottom of the foundation may also be assumed to be replaced by an equivalent surcharge, q = γDf (where γ = unit weight of soil). The failure zone under the foundation can be separated into three parts (see figure 3.5).

1. The triangular zone ACD immediately under the foundation 2. The radial shear zones ADF and CDE, with the curves DE and DF being arcs of

logarithmic spiral 3. Two triangular Rankine passive zone AFH and CEG

The angles CAD and ACD are assumed to be equal to the soil friction angle, ϕ. Note that, with the replacement of the soil above the bottom of the foundation by an equivalent surcharge q, the shear resistance of the soil along the failure surfaces GI and HJ was neglected.

Using the equilibrium analysis, Terzaghi expressed the ultimate bearing capacity in the form

qu = cNc + qNq + 12γBNγ (strip foundation) [3.3]

Where

c = cohesion of soil

γ = unit weight of soil

q = γDf

Nc, Nq, Nγ =bearing capacity factors that are nondimensional and are only functions of the soil friction angle,ϕ

The bearing capacity factors Nc, Nq, and Nγ are defined by

Nc = cotϕ�e2(3π/4−ϕ2)tan ϕ

2 cos 2�π4+ϕ2�

− 1� = cotϕ(Nq − 1) [3.4]

Nq = e2(3π/4−ϕ2)tan ϕ

2 cos 2�45+ϕ2�

[3.5]

Nγ = 12� Kpγ

cos 2 ϕ− 1� tanϕ [3.6]


Where Kpγ = passive pressure coefficient

The variations of the bearing capacity factors defined by equations (4, 5, and 6) are given in table 1,

Table 1 Terzaghi’s Bearing Capacity Factors-equations (4, 5, and 6)

ϕ Nc Nq Nγ ϕ Nc Nq Nγ

0 5.70 1.00 0.00 26 27.09 14.21 9.84

1 6.00 1.1 0.01 27 29.24 15.90 11.60

2 6.30 1.22 0.04 28 31.61 17.81 13.70

3 6.62 1.35 0.06 29 34.24 19.98 16.18

4 6.97 1.49 0.10 30 37.16 22.46 19.13

5 7.34 1.64 0.14 31 40.41 25.28 22.65

6 7.73 1.81 0.20 32 44.04 28.52 26.87

7 8.15 2.00 0.27 33 48.09 32.23 31.94

8 8.60 2.21 0.35 34 52.64 36.50 38.04

9 9.09 2.44 0.44 35 57.75 41.44 45.41

10 9.61 2.69 0.56 36 63.53 47.16 54.36

11 10.16 2.98 0.69 37 70.01 53.80 65.27

12 10.76 3.29 0.85 38 77.50 61.55 78.61

13 11.41 3.63 1.04 39 85.97 70.61 95.03

14 12.11 4.02 1.26 40 95.66 81.27 115.31

15 12.86 4.45 1.52 41 106.81 93.85 140.51

16 13.68 4.92 1.82 42 119.67 108.75 171.99

17 14.60 5.45 2.18 43 134.58 126.50 211.56

18 15.12 6.04 2.59 44 151.95 147.74 261.60


19 16.56 6.70 3.07 45 172.28 173.28 325.34

20 17.69 7.44 3.64 46 196.22 204.19 407.11

21 18.92 8.26 4.31 47 224.55 241.80 512.84

22 20.27 9.19 5.09 48 258.28 287.85 650.67

23 21.75 10.23 6.00 49 298.71 344.63 831.99

24 23.36 11.40 7.08 50 347.50 415.14 1072.80

25 25.13 12.72 8.34

From Kumbhojkar (`1993)

For estimating the ultimate bearing capacity of square or circular foundations equation (1) may be modified to

qu = 1.3cNc + qNq + 0.4γBNγ (square foundation) [3.7]

And

qu = 1.3cNc + qNq + 0.3γBNγ (circular foundation) [3.8]

In equation (7), B equals the dimension of each side of the foundation; in equation (8), B equals the diameter of the foundation.

For foundations that exhibit the local shear failure mode in soils, Terzaghi suggested modifications to equations (3, 7, and 8) as follows:

qu = 23

cN′c + qN′q + 12γBN′γ (strip foundation) [3.9]

qu = 0.867cN′c + qN′q + 0.4γBN′γ (square foundation) [3.10]

qu = 0.867cN′c + qN′q + 0.3γBN′γ (circular foundation) [3.11]

N′c , N′q , and N′γ are the modified bearing capacity factors. They can be calculated by using the bearing capacity factor equations (for Nc, Nq, and Nγ) by replacing ϕ by ϕ′ = tan−1(2

3 tanϕ). the variation of N′c , N′q , and N′γ with the soil friction angle, ϕ, is given in table 2.

Terzaghi’s bearing capacity equations have now been modified to take into account the effects of the foundation shape (B/L), depth of embedment (Df), and the load inclination. This is given in section 7. Many design engineers, however, still use Terzaghi’s equation,


which provides fairly good results considering the uncertainty of the soil conditions at various sites.

MODIFICATION OF BEARING CAPACITY EQUATIONS FOR WATER TABLE

Equations (3) and (7) to (11) have been developed for determining the ultimate bearing capacity based on the assumption that the water table is located well below the foundation. However, if the water table is close to the foundation, some modifications of the bearing capacity equations will be necessary, depending on the location of the water table (see figure 3.6).

Figure 3.6 Modification of bearing capacity equations for water table

Case I

If the water table is located so that 0 ≤ D1 ≤ Df, the factor q in the bearing capacity equations takes the form

Table 2 Terzaghi’s Modified Bearing Capacity Factors Nc, Nq, and Nγ

ϕ N′c N′q Nγ ′ ϕ N′c N′q Nγ

′

0 5.70 1.00 0.00 26 15.53 6.05 2.59

1 5.90 1.07 0.005 27 16.30 6.54 2.88

2 6.10 1.14 0.02 28 17.13 7.07 3.29

3 6.30 1.22 0.04 29 18.03 7.66 3.76

4 6.51 1.30 0.055 30 18.99 8.31 4.39


5 6.74 1.39 0.074 31 20.03 9.03 4.83

6 6.97 1.49 0.10 32 21.16 9.82 5.51

7 7.22 1.59 0.128 33 22.39 10.69 6.32

8 7.47 1.70 0.16 34 23.72 11.67 7.22

9 7.74 1.82 0.20 35 25.18 12.75 8.35

10 8.02 1.94 0.24 36 26.77 13.97 9.41

11 8.32 2.08 0.30 37 28.51 15.32 10.90

12 8.63 2.22 0.35 38 30.43 16.85 12.75

13 8.96 2.38 0.42 39 32.53 18.56 14.71

14 9.31 2.55 0.48 40 34.87 20.50 17.22

15 9.67 2.73 0.57 41 37.45 22.70 19.75

16 10.06 2.92 0.67 42 40.33 25.21 22.50

17 10.47 3.13 0.76 43 43.54 28.06 26.25

18 10.90 3.36 0.88 44 47.13 31.34 30.40

19 11.36 3.61 1.03 45 51.17 35.11 36.00

20 11.85 3.88 1.12 46 55.73 39.48 41.70

21 12.37 4.17 1.35 47 60.91 44.45 49.30

22 12.92 4.48 1.55 48 66.80 50.46 59.25

23 13.51 4.82 1.74 49 73.55 57.41 71.45

24 14.14 5.20 1.97 50 81.31 65.60 85.75

25 14.80 5.60 2.25

𝑞𝑞 = 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒ℎ𝑎𝑎𝑠𝑠𝑎𝑎𝑒𝑒 = 𝐷𝐷1𝛾𝛾 + 𝐷𝐷2(𝛾𝛾𝑠𝑠𝑎𝑎𝑒𝑒 − 𝛾𝛾𝑤𝑤) [3.12]

Where

𝛾𝛾𝑠𝑠𝑎𝑎𝑒𝑒 = 𝑠𝑠𝑎𝑎𝑒𝑒𝑠𝑠𝑠𝑠𝑎𝑎𝑒𝑒𝑒𝑒𝑠𝑠 𝑠𝑠𝑢𝑢𝑒𝑒𝑒𝑒 𝑤𝑤𝑒𝑒𝑒𝑒𝑎𝑎ℎ𝑒𝑒 𝑜𝑜𝑒𝑒 𝑠𝑠𝑜𝑜𝑒𝑒𝑠𝑠


𝛾𝛾𝑤𝑤 = 𝑠𝑠𝑢𝑢𝑒𝑒𝑒𝑒 𝑤𝑤𝑒𝑒𝑒𝑒𝑎𝑎ℎ𝑒𝑒 𝑜𝑜𝑒𝑒 𝑤𝑤𝑎𝑎𝑒𝑒𝑒𝑒𝑠𝑠

Also, the value of 𝛾𝛾 in the last term of the equations has to be replaced by 𝛾𝛾 ′ = 𝛾𝛾𝑠𝑠𝑎𝑎𝑒𝑒 − 𝛾𝛾𝑤𝑤 .

Case II

For a water table located so that 0 ≤ 𝑠𝑠 ≤ 𝐵𝐵,

𝑞𝑞 = 𝛾𝛾𝐷𝐷𝑒𝑒 [3.13]

The factor 𝛾𝛾 in the last term of the bearing capacity equations must be replaced by the factor

�̅�𝛾 = 𝛾𝛾 ′ + 𝑠𝑠𝐵𝐵

(𝛾𝛾 − 𝛾𝛾 ′) [3.14]

The preceding modifications are based on the assumption that there is no seepage force in the soil.

Case III

When he water table is located so that 𝑠𝑠 ≤ 𝐵𝐵,the water will have no effect on the ultimate bearing capacity.

CASE HISTORY: ULTIMATE BEARING CAPACITY IN SATURATED CLAY

Brand et al. (1972) reported field test results for small foundations on soft Bangkok clay (a deposit of marine clay) in Rangsit, Thailand. The results of the soil exploration are shown in figure 3.7. Because of the sensitivity of the clay, the laboratory test results for 𝑒𝑒𝑠𝑠 (uconfined compression and unconsolidated undrianed triaxial) were rather scattered; however, they obtained better results for the variation of 𝑒𝑒𝑠𝑠 with depth from field vane shear tests. The vane shear test results showed that the average variations of the undrained cohesion were

Depth (m) 𝑒𝑒𝑠𝑠(𝑘𝑘𝑘𝑘/𝑚𝑚2)

0-1.5 ≈ 35

1.5-2 Decreasing linearly from 35 to 24

2-8 ≈ 24


Five small square foundations were tested for ultimate bearing capacity. The sizes of the foundations were, 0.6 m,× 0.6 m, 0.675 m × 0.75 m × 0.75m, 0.9 m × 0.9 m, and 1.05 × 1.05 m. The depth of the bottom of the foundations was 1.5 m measured from the ground surface. The load-settlement plots obtained from the bearing capacity tests are shown in figure 3.8.

Figure 3.7 Results of soil exploration in soft Bangkok clay at Rangsit. Thailand (after Brand et al., 1972)

Figure 3.8 Loan-settlement plots obtained from bearing capacity tests


Analysis of the Field Test Results

The ultimate loads, 𝑄𝑄𝑠𝑠 , obtained from each test are also shown in figure 3.8. The ultimate load is defined as the point where the load displacement becomes practically linear. The failure in soil below the foundation is of local shear type.

Hence, from equation (10)

𝑞𝑞𝑠𝑠 = 0.867𝑒𝑒𝑠𝑠𝑘𝑘′𝑒𝑒 + 𝑞𝑞𝑘𝑘′𝑞𝑞 + 0.4𝛾𝛾𝐵𝐵𝑘𝑘′𝛾𝛾

For 𝜙𝜙 = 0, 𝑒𝑒 = 𝑒𝑒𝑠𝑠 𝑎𝑎𝑢𝑢𝑠𝑠,𝑒𝑒𝑠𝑠𝑜𝑜𝑚𝑚 𝑒𝑒𝑎𝑎𝑡𝑡𝑠𝑠𝑒𝑒 2,𝑘𝑘′𝑒𝑒 = 5.7,𝑘𝑘′

𝑞𝑞 = 1 𝑎𝑎𝑢𝑢𝑠𝑠 𝑘𝑘′𝛾𝛾 = 0. Thus for 𝜙𝜙 = 0

𝑞𝑞𝑠𝑠 = 4.94𝑒𝑒𝑠𝑠 + 𝑞𝑞 [3.15]

If we assume that the unit weight of soil is about 18.5 𝑘𝑘𝑘𝑘/𝑚𝑚3, 𝑞𝑞 ≈ 𝐷𝐷𝑒𝑒𝛾𝛾 = (1.5)(18.5) = 27.75 𝑘𝑘𝑘𝑘/𝑚𝑚2. We can then assume average values of 𝑒𝑒𝑠𝑠 : for depths of 1.5 m to 2.0 m, 𝑒𝑒𝑠𝑠 ≈ (35 + 24)/2 = 29.5 𝑘𝑘𝑘𝑘/𝑚𝑚2; for depths greater than 2.0 𝑚𝑚, 𝑒𝑒𝑠𝑠 ≈ (24 𝑘𝑘𝑘𝑘/𝑚𝑚2. If we assume that the undrained cohesion of clay at depth ≤ 𝐵𝐵 below the foundation controls the ultimate bearing capacity,

Table 3 Comparison of Theoretical and Field Ultimate Bearing Capacities

(𝐵𝐵)𝑚𝑚 𝐶𝐶𝑠𝑠(𝑎𝑎𝑒𝑒𝑒𝑒𝑠𝑠𝑎𝑎𝑎𝑎𝑒𝑒 )𝑎𝑎 Plasticity 𝑒𝑒𝑢𝑢𝑠𝑠𝑒𝑒𝑖𝑖𝑡𝑡

Correlation factor, 𝜆𝜆𝑒𝑒

𝐶𝐶𝑠𝑠(𝑒𝑒𝑜𝑜𝑠𝑠𝑠𝑠𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑠𝑠 )𝑠𝑠 (𝑘𝑘𝑘𝑘/𝑚𝑚2)

𝑞𝑞𝑠𝑠(𝑒𝑒ℎ𝑒𝑒𝑜𝑜𝑠𝑠𝑒𝑒 )𝑎𝑎(𝑘𝑘𝑘𝑘

/𝑚𝑚2 𝑄𝑄𝑠𝑠(𝑒𝑒𝑒𝑒𝑒𝑒𝑠𝑠𝑠𝑠 )

𝑒𝑒(𝑘𝑘𝑘𝑘) 𝑄𝑄𝑠𝑠(𝑒𝑒𝑒𝑒𝑒𝑒𝑠𝑠𝑠𝑠 )𝑞𝑞(𝑘𝑘𝑘𝑘

/𝑚𝑚2)

0.6 28.58 40 0.84 24.01 146.4 60 166.6

0.675 28.07 40 0.84 23.58 144.2 71 155.8

0.75 27.67 40 0.84 23.24 142.6 90 160

0.9 27.06 40 0.84 22.73 140.0 124 153

1.05 26.62 40 0.84 22.36 138.2 140 127

a Equation (16)

b From figure 3.7

c From table 7 [𝜆𝜆 = 1.7 − 0.54 𝑠𝑠𝑜𝑜𝑎𝑎(𝑃𝑃𝑃𝑃):𝐵𝐵𝐵𝐵𝑒𝑒𝑠𝑠𝑠𝑠𝑠𝑠𝑚𝑚 (1972)]

d Equation (19 from chapter 2)


e Equation (15)

f Figure 3.8

g 𝑄𝑄𝑠𝑠(𝑒𝑒𝑒𝑒𝑒𝑒𝑠𝑠𝑠𝑠 )/𝐵𝐵2

𝑒𝑒𝑠𝑠(𝑎𝑎𝑒𝑒𝑒𝑒𝑠𝑠𝑎𝑎𝑎𝑎𝑒𝑒 ) ≈(29.5)(2.0−1.5)+(24)[𝐵𝐵−(2.0−1.5)]

𝐵𝐵 [3.16]

The 𝑒𝑒𝑠𝑠(𝑎𝑎𝑒𝑒𝑒𝑒𝑠𝑠𝑎𝑎𝑎𝑎𝑒𝑒 ) value obtained for each foundation needs to be corrected in view of equation (19 from chapter 2) table 3 presents the details of other calculations and a comparison of the theoretical and field ultimate bearing capacities.

Note that the ultimate bearing capacities obtained from the field are about 10% higher than those obtained from theory. One reason for such a difference is that the ratio 𝐷𝐷𝑒𝑒/𝐵𝐵 for the field tests varies from 1.5 to 2.5. The increase of the bearing capacity due to the depth of embedment has not been accounted for in equation (16).

FACTOR SAFETY

Calculating the gross allowable load-bearing capacity of shallow foundations requires application of a factor of safety (FS) to the gross ultimate bearing capacity, or

𝑞𝑞𝑎𝑎𝑠𝑠𝑠𝑠 = 𝑞𝑞𝑠𝑠𝐹𝐹𝐹𝐹

[3.17]

However, some practicing engineers prefer to use a factor of safety of

𝑘𝑘𝑒𝑒𝑒𝑒 𝑠𝑠𝑒𝑒𝑠𝑠𝑒𝑒𝑠𝑠𝑠𝑠 𝑒𝑒𝑢𝑢𝑒𝑒𝑠𝑠𝑒𝑒𝑎𝑎𝑠𝑠𝑒𝑒 𝑜𝑜𝑢𝑢 𝑠𝑠𝑜𝑜𝑒𝑒𝑠𝑠 = 𝑢𝑢𝑒𝑒𝑒𝑒 𝑠𝑠𝑠𝑠𝑒𝑒𝑒𝑒𝑚𝑚𝑎𝑎𝑒𝑒𝑒𝑒 𝑡𝑡𝑒𝑒𝑎𝑎𝑠𝑠𝑒𝑒𝑢𝑢𝑎𝑎 𝑒𝑒𝑎𝑎𝑐𝑐𝑎𝑎𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝐹𝐹𝐹𝐹

[3.18]

The net ultimate bearing capacity is defined as the ultimate pressure per unit area of the foundation that can be supported by the soil in excess of the pressure caused by the surrounding soil at the foundation level. If the difference between the unit weights of concrete used in the foundation and the unit weight of soil surrounding is assumed to be negligible,

𝑞𝑞𝑢𝑢𝑒𝑒𝑒𝑒 (𝑠𝑠) = 𝑞𝑞𝑠𝑠 − 𝑞𝑞 [3.19]

Where

𝑞𝑞𝑢𝑢𝑒𝑒𝑒𝑒 (𝑠𝑠) = 𝑢𝑢𝑒𝑒𝑒𝑒 𝑠𝑠𝑠𝑠𝑒𝑒𝑒𝑒𝑚𝑚𝑎𝑎𝑒𝑒𝑒𝑒 𝑡𝑡𝑒𝑒𝑎𝑎𝑠𝑠𝑒𝑒𝑢𝑢𝑎𝑎 𝑒𝑒𝑎𝑎𝑐𝑐𝑎𝑎𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒

𝑞𝑞 = 𝛾𝛾𝐷𝐷𝑒𝑒

So,


𝑄𝑄𝑎𝑎𝑠𝑠𝑠𝑠 (𝑢𝑢𝑒𝑒𝑒𝑒 ) = 𝑞𝑞𝑠𝑠−𝑞𝑞𝐹𝐹𝐹𝐹

[3.20]

The factor of safety as defined by equation (20) may be at least 3 in all cases.

Another type of factor of safety for the bearing capacity of shallow foundations is often used. It is the factor with respect to shear failure (FSshear ). in most cases, a vale of FSshear = 1.4 − 1.6 is desirable along with a minimum factor of safety of 3-4 against gross or net ultimate bearing capacity. The following procedure should be used to calculate the net allowable load for a given FSshear .

1. Let c and ϕ be the cohesion and the angle of friction, respectively, of soil and let FSshear be the required factor of safety with respect to shear failure. So the developed cohesion and the angle of friction are cd = c

FS sh ear [3.21]

ϕd = tan−1 � tan ϕFS sh ear

� [3.22]

2. The gross allowable bearing capacity can now be calculated according to equations (3, 7, 8), with cd and ϕd as the shear strength parameters of the soil. For example, the gross allowable bearing capacity of a continuous foundation according to Terzaghi’s equation is qall = cd Nc + qNq + 1

2γBNγ [3.23] Where Nc, Nq, and Nγ = bearing capacity factors for the friction angle, ϕd

3. The net allowable bearing capacity is thus qall (net ) = qall − q = cd Nc + q�Nq − 1� + 1

2γBNγ [3.24]

Irrespective of the procedure by which the factor of safety is applied, the magnitude of FS should depend on the uncertainties and risks involved for the conditions encountered.

Example 1

A square foundation is 5 ft × 5 ft in plan. The soil supporting the foundation has a friction angle of ϕ = 20° and c = 320 lb/ft2. The unit weight of soil, γ, is 115 lb/ft3. Determine the allowable gross load on the foundation with a factor of safety (FS) of 4. Assume that the depth of the foundation (Df) is 3 ft and that general shear failure occurs in the soil.


Solution

From equation (7)

qu = 1.3cNc + qNq + 0.4γBNγ

From table 1, for ϕ = 20°,

Nc = 17.69

Nq = 7.44

Nγ = 3.64

Thus

qu = (1.3)(320)(17.69) + (3 × 115)(7.44) + (0.4)(115)(5)(3.64)

= 7359 + 2567 + 837 = 10, 736 lb/ft2

So, the allowable load per unit area of the foundation is

qall = quFS

= 10,7364

≈ 2691 lb/ft2

Thus load total allowable gross load is

Q = (2691)B2 = (2691)(5 × 5) = 67,275 lb

topicsnptel.ac.in/courses/105104137/module3/lecture9.pdfnptel – advanced foundation engineering-i...

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