CE 333 Geotechnical Engineering II Sultan Mohammad Farooq Sheikh Sharif Ahmed Department of Civil Engineering Chittagong University of Engineering & Technology Bearing Capacity of Shallow Foundation οΆ In several instances, as with the base of a retaining wall, foundations are subjected to moments in addition to the vertical load, as shown in Figure 08a. οΆ In such cases, the distribution of pressure by the foundation on the soil is not uniform. Bearing Capacity of Shallow Foundation Figure 08 Nature of Pressure Distribution for Eccentric Loading Bearing Capacity of Shallow Foundation The nominal distribution of pressure isππππ πΈ ππ΄ = + π π©π³ π© π³ ππππ πΈ ππ΄ = − π π©π³ π© π³ and Where, πΈ = π‘ππ‘ππ π£πππ‘ππππ ππππ π΄ = ππππππ‘ ππ π‘βπ πππ’ππππ‘πππ Bearing Capacity of Shallow Foundation Figure 08b shows a force system equivalent to that shown in Figure 09a. The distanceπ΄ π= πΈ is the eccentricity. Substituting Eq. (3.35) into Eqs. (3.33) and (3.34) gives πΈ ππ ππππ = π+ π©π³ π© and πΈ ππ ππππ = π− π©π³ π© Bearing Capacity of Shallow Foundation Figure 08 Nature of Pressure Distribution for Eccentric Loading Bearing Capacity of Shallow Foundation οΆ Note that, in these equations, when the eccentricity e becomes π© π , ππππ is zero. For π > π© π , ππππ will be negative, which means that tension will develop. οΆ Because soil cannot take any tension, there will then be a separation between the foundation and the soil underlying it. οΆ The nature of the pressure distribution on the soil will be as shown in Figure 08a. The value of ππππ is then ππΈ ππππ = ππ³ π© − ππ οΆ The exact distribution of pressure is difficult to estimate. Bearing Capacity of Shallow Foundation οΆ Figure 09 shows the nature of failure surface in soil for a surface strip foundation subjected to an eccentric load. οΆ The factor of safety for such type of loading against bearing capacity failure can be evaluated asπΈπππ ππΊ = πΈ Where, ππ’ππ‘ = π’ππ‘ππππ‘π ππππ πππππ¦πππ πππππππ‘π¦ Bearing Capacity of Shallow Foundation Figure 09 Failure surface for a surface strip foundation subjected to an eccentric load Bearing Capacity of Shallow Foundation Figure 10 One-Way Eccentricity Bearing Capacity of Shallow Foundation Effective Area Method (Meyerhoff, 1953) οΆ In 1953, Meyerhof proposed a theory that is generally referred to as the effective area method. οΆ The following is a step-by-step procedure for determining the ultimate load that the soil can support and the factor of safety against bearing capacity failure: Bearing Capacity of Shallow Foundation STEP 1 οΆ Determine the effective dimensions of foundation π©′ = πππππππππ πππ ππ = π© − ππ π³′ = πππππππππ ππππππ = π³ the οΆ Note that if the eccentricity were in the direction of the length of the foundation, the value of π³′ would be equal to π³ − ππ. The value of π©′ would equal B. οΆ The smaller of the two dimensions (i.e., π³′ πππ π©′ ) is the effective width of the foundation. Bearing Capacity of Shallow Foundation STEP 2 οΆ Use the following equation for the ultimate bearing capacity: π ππ = ππ΅π πππ πππ πππ + ππ΅π πππ πππ πππ + πΈπ©′π΅πΈ ππΈπ ππΈπ ππΈπ π οΆ To evaluate πππ , πππ and ππΈπ , use the relationships given in Table 03 with effective length and effective width dimensions instead of L and B, respectively. οΆ To determine πππ , πππ and ππΈπ , use the relationships given in Table 03. However, do not replace B with π©′ . Bearing Capacity of Shallow Foundation STEP 3 οΆ The total ultimate load that the foundation can sustain is π¨′ πΈπππ = ππ (π©′)(π³′) Where, π¨′ = πππππππππ ππππ Bearing Capacity of Shallow Foundation STEP 4 οΆ The factor of safety against bearing capacity failure is- πΈπππ ππΊ = πΈ Bearing Capacity of Shallow Foundation ∅ Nc Nq π΅πΈ (M) ∅ Nc Nq π΅πΈ (M) 0° 1° 2° 3° 4° 5° 6° 7° 8° 9° 10° 11° 12° 13° 14° 15° 16° 17° 18° 19° 20° 21° 22° 23° 24° 5.10 5.38 5.63 5.90 6.19 6.49 6.81 7.16 7.53 7.92 8.34 8.80 9.28 9.81 10.37 10.98 11.63 12.34 13.10 13.93 14.83 15.81 16.88 18.05 19.32 1.00 1.09 1.20 1.31 1.43 1.57 1.72 1.88 2.06 2.25 2.47 2.71 2.97 3.26 3.59 3.94 4.34 4.77 5.26 5.80 6.40 7.07 7.82 8.66 9.60 0.00 0.00 0.01 0.02 0.04 0.07 0.11 0.15 0.21 0.28 0.37 0.47 0.60 0.74 0.92 1.13 1.37 1.66 2.00 2.40 2.87 3.42 4.07 4.82 5.72 25° 26° 27° 28° 29° 30° 31° 32° 33° 34° 35° 36° 37° 38° 39° 40° 41° 42° 43° 44° 45° 46° 47° 48° 49° 20.72 22.25 23.94 25.80 27.86 30.14 32.67 35.49 38.64 42.16 46.12 50.59 55.63 61.35 67.87 75.31 83.86 93.71 105.11 118.37 133.87 152.10 173.64 199.26 229.93 10.66 11.85 13.20 14.72 16.44 18.40 20.63 23.18 26.09 29.44 33.30 37.75 42.92 48.93 55.96 64.20 73.90 85.37 99.01 115.31 134.87 158.50 187.21 222.30 265.50 6.77 8.00 9.46 11.19 13.24 15.67 18.56 22.02 26.17 31.15 37.15 44.43 53.27 64.07 77.33 93.69 113.99 139.32 171.14 211.41 262.74 328.73 414.33 526.46 674.92 Bearing Capacity of Shallow Foundation Author Factor Condition πππ ∅ = 0° Relationship πΉππ = 1 + 0.2 πΉππ = πΉπΎπ = 1.0 Shape πππ ∅ ≥ 10° πΉππ = 1 + 0.2 π΅ πΏ Meyerhof πΉππ = πΉπΎπ = 1 + 0.1 πππ ∅ = 0° π‘ππ2 45 + π΅ πΏ ∅ 2 π‘ππ2 45 + πΉππ = 1 + 0.2 ∅ 2 π·π π΅ πΉππ = πΉπΎπ = 1.0 Depth πππ ∅ ≥ 10° πΉππ = 1 + 0.2 π·π πππ πππ¦ ∅ π‘ππ 45 + π΅ πΉππ = πΉπΎπ = 1 + 0.1 Inclination π΅ πΏ π·π π΅ πΉππ = πΉππ = 1 − π‘ππ 45 + πΌ° 2 90° πΌ° 2 ∅° πππ ∅ > 0° πΉπΎπ = 1 − πππ ∅ = 0° πΉπΎπ = 0 ∅ 2 ∅ 2 Bearing Capacity of Shallow Foundation Prakash and Saran Theory οΆ Prakash and Saran (1971) analyzed the problem of ultimate bearing capacity of eccentrically and vertically loaded continuous (strip) foundations by using the one-sided failure surface in soil, as shown in Figure 09. οΆ According to this theory, the ultimate load per unit length of a continuous foundation can be estimated as π πΈπππ = π© ππ΅π(π) + ππ΅π(π) + πΈπ©π΅πΈ(π) π Where, ππ(π) , ππ(π) , ππΎ(π) = πππππππ πππππππ‘π¦ ππππ‘πππ π’ππππ ππππππ‘πππ πππππππ Bearing Capacity of Shallow Foundation Figure 09 Failure surface for a surface strip foundation subjected to an eccentric load Bearing Capacity of Shallow Foundation Prakash and Saran Theory οΆ The variations of π΅π(π) , π΅π(π) and π΅πΈ(π) with soil friction angle ∅ are given in Figures 11, 12, and 13. οΆ For rectangular foundations, the ultimate load can be given as πΈπππ π = π©π³ ππ΅π(π) πππ(π) + ππ΅π(π) πππ(π) + πΈπ©π΅πΈ(π) ππΈπ(π) π Where, πΉππ (π) , πΉππ (π) and πΉπΎπ (π) = π βπππ ππππ‘πππ Bearing Capacity of Shallow Foundation Figure 11 Variations of π΅π(π) Bearing Capacity of Shallow Foundation Figure 12 Variations of π΅π(π) Bearing Capacity of Shallow Foundation Figure13 Variations of π΅πΈ(π) Bearing Capacity of Shallow Foundation Prakash and Saran Theory Prakash and Saran (1971) also recommended the following for the shape factors: π³ πππ(π) = π. π − π. πππ (ππππ π πππππππ ππ π. π) π© πππ π = π. π ππ π© π π π© ππΈπ(π) = π. π + − π. ππ + π. ππ − π© π³ π π© π³ π Bearing Capacity of Shallow Foundation Reduction Factor Method (For Granular Soil) Purkayastha and Char (1977) carried out stability analysis of eccentrically loaded continuous foundations supported by a layer of sand using the method of slices. Based on that analysis, they proposedππ(ππππππππ) πΉπ = π − ππ(πππππππ) Where, π π = ππππ’ππ‘πππ ππππ‘ππ ππ’(πππππ‘πππ) = π’ππ‘ππππ‘π πππππππ πππππππ‘π¦ ππ ππππππ‘πππππππ¦ ππππππ ππππ‘πππ’ππ’π πππ’ππππ‘ππππ ππ’ ππππ‘πππ = π’ππ‘ππππ‘π πππππππ πππππππ‘π¦ ππ ππππ‘πππππ¦ ππππππ ππππ‘πππ’ππ’π πππ’ππππ‘ππππ Bearing Capacity of Shallow Foundation Reduction Factor Method (For Granular Soil) The magnitude of πΉπ can be expressed asπ π πΉπ = π π© Where, a and k are functions of the embedment ratio π«π π© Table 04 Variations of a and k π«π π© 0.00 0.25 0.50 1.00 a k 1.862 1.811 1.754 1.820 0.730 0.785 0.800 0.888 Bearing Capacity of Shallow Foundation Reduction Factor Method (For Granular Soil) Hence, combining ππ(ππππππππ) = ππ(πππππππ) π − πΉπ = ππ(πππππππ) π π−π π© π Where, π ππ(πππππππ) = ππ΅π πππ + πΈπ©π΅πΈ ππΈπ π The relationships for πππ and ππΈπ are given in Table 03. The ultimate load per unit length of the foundation can then be given as πΈπ = π©ππ(ππππππππ)
