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 Consider a situation in which a foundation is subjected to a vertical ultimate load πΈπππ and a moment M, as shown in Figures 14a and b. οΆ For this case, the components of the moment M about the x-axes and y-axes can be determined as π΄π and π΄π , respectively (See Figure 14c). οΆ This condition is equivalent to a load πΈπππ placed eccentrically on the foundation with π = ππ© πππ π = ππ³ (Figure 14d). Note that π΄π ππ© = πΈπππ and π΄π ππ³ = πΈπππ Bearing Capacity of Shallow Foundation Figure 14 Two-Way Eccentricity Bearing Capacity of Shallow Foundation οΆ If πΈπππ is needed, it can be obtained from Where, πΈπππ = ππ π¨′ π ππ = ππ΅π πππ πππ πππ + ππ΅π πππ πππ πππ + πΈπ©′π΅πΈ ππΈπ ππΈπ ππΈπ π ′ π¨ = πππππππππ ππππ = π©′π³′ οΆ As before, to evaluate πππ , πππ , and ππΈπ (Table 03), we use the effective length π³′ and effective width π©′ instead of L and B, respectively. οΆ To calculate πππ , πππ , and ππΈπ we do not replace B with π©′ . οΆ In determining the effective area π¨′ , effective width π©′ and effective length π³′ , five possible cases may arise (Highter and Anders, 1985). Bearing Capacity of Shallow Foundation ∅ Nc Nq π΅πΈ (M) π΅πΈ (V) π΅πΈ (H) ∅ Nc Nq π΅πΈ (M) π΅πΈ (V) π΅πΈ (H) 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 0.00 0.07 0.15 0.24 0.34 0.45 0.57 0.71 0.86 1.03 1.22 1.44 1.69 1.97 2.29 2.65 3.06 3.53 4.07 4.68 5.39 6.20 7.13 8.20 9.44 0.00 0.00 0.01 0.02 0.05 0.07 0.11 0.16 0.22 0.30 0.39 0.50 0.63 0.78 0.97 1.18 1.43 1.73 2.08 2.48 2.95 3.50 4.13 4.88 5.75 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 10.88 12.54 14.47 16.72 19.34 22.40 25.99 30.21 35.19 41.06 48.03 56.31 66.19 78.02 92.25 109.41 130.21 155.54 186.53 224.64 271.75 330.34 403.66 496.00 613.15 6.76 7.94 9.32 10.94 12.84 15.07 17.69 20.79 24.44 28.77 33.92 40.05 47.38 56.17 66.76 79.54 95.05 113.96 137.10 165.58 200.81 244.65 299.52 368.67 456.41 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 Author Factor Condition Relationship πΉππ = 1 + Shape πππ πππ ∅ ππ π΅ πΏ ππ π΅ πΉππ = 1 + tan ∅ πΏ π΅ πΉπΎπ = 1 − 0.4 πΏ Hansen & Vesic π·π π·π πππ ≤ 1.0 π΅ πΉππ = 1 + 0.4 πππ ∅ = 0° π΅ 1 − πΉππ πΉππ = πΉππ − πππ ∅ > 0° ππ tan ∅ πππ πππ ∅ π·π πΉππ = 1 + 2 tan ∅ 1 − π ππ∅ 2 π΅ πΉπΎπ = 1.0 Depth π·π πππ > 1.0 π΅ & πππ πππ ∅ πΉππ = 1 + 0.4 π‘ππ−1 πΉππ = 1 + 2 tan ∅ 1 − π ππ∅ 2 π·π π΅ π‘ππ −1 πΉπΎπ = 1.0 Note: πππ−π π«π π© is in radians π·π π΅ Bearing Capacity of Shallow Foundation Hansen Author Factor Condition Relationship πππ ∅ > 0° 1 − πΉππ πΉππ = πΉππ − ππ − 1 πππ ∅ = 0° Inclination ππ» πΉππ = 0.5 1 − π΄π ππ πΉππ = 1 − πππ πππ¦ ∅ 0.5 0.5ππ» ππ +π΄π ππ cot ∅ 5 0.7ππ» πΉπΎπ = 1 − ππ + π΄π ππ cot ∅ 5 Bearing Capacity of Shallow Foundation Author Factor Condition Vesic πππ ∅ > 0° πππ ∅ = 0° Inclination πππ πππ¦ ∅ Relationship πΉππ = πΉππ − 1 − πΉππ ππ − 1 πππ» πΉππ = 1 − π΄π ππ ππ 0.5 ππ» πΉππ = 1 − ππ + π΄π ππ cot ∅ ππ» πΉπΎπ = 1 − ππ + π΄π ππ cot ∅ π π+1 Bearing Capacity of Shallow Foundation CASE I ππ³ π ππ© π ≥ πππ ≥ π³ π π© π The effective area for this condition is shown in Figure 15, or π¨′ Where, And π = π©π π³ π π πππ© π©π = π© π. π − π© πππ³ π³π = π³ π. π − π³ The effective length is the larger of the two dimensions π©π πππ π³π . So the effective width is π©′ π¨′ = π³′ Bearing Capacity of Shallow Foundation Figure 15 Effective Area (Case I) Bearing Capacity of Shallow Foundation CASE II ππ³ ππ© π < π. π πππ π < < π³ π© π οΆ The effective area for this case, shown in Figure 16, is π¨′ π = π³π + π³π π© π οΆ The magnitudes of π³π πππ π³π can be determined from Figure 17. οΆ The effective length is π³′ = π³π ππ π³π (whichever is larger) οΆ The effective width is π©′ = π¨′ π³′ Bearing Capacity of Shallow Foundation Figure 16 Effective Area (Case II) Bearing Capacity of Shallow Foundation Figure 17 Chart for Obtaining π³π π³ πππ π³π π³ Bearing Capacity of Shallow Foundation CASE III ππ³ π ππ© < πππ π < < π. π π³ π π© οΆ The effective area, shown in Figure 18, is π¨′ π = π©π + π©π π³ π οΆ The effective width is οΆ The effective length is ′ π¨ π©′ = π³ π³′ = π³ οΆ The magnitudes of π©π πππ π©π can be determined from Figure 19. Bearing Capacity of Shallow Foundation Figure 18 Effective Area (Case III) Bearing Capacity of Shallow Foundation Figure 19 Chart for Obtaining π©π π© πππ π©π π© Bearing Capacity of Shallow Foundation CASE IV ππ³ π ππ© π < πππ < π³ π π© π οΆ Figure 20 shows the effective area for this case. οΆ The ratio π©π π© and thus π©π , can be determined by using the ππ³ π³ curves (Figure 21) that slope upward. οΆ Similarly, the ratio π³π π³ and thus π³π , can be determined by using the ππ³ π³ curves (Figure 21) that slope downward. The effective area is thenπ¨′ π = π³ π π© + π© + π©π π³ − π³ π π π©′ π¨′ π³ The effective width is = The effective length is π³′ = π³ Bearing Capacity of Shallow Foundation Figure 20 Effective Area (Case IV) Bearing Capacity of Shallow Foundation Figure 21 Chart for Obtaining π©π π© πππ π³π π³ Bearing Capacity of Shallow Foundation CASE V Circular Foundation οΆ In the case of circular foundations under eccentric loading (Figure 22), the eccentricity is always one way. οΆ The effective area π¨′ and the effective width π©′ for a circular foundation are given in a non-dimensional form in Table 05 and in Figure 23. οΆ Once π¨′ and π©′ are determined, the effective length can be obtained asπ³′ π¨′ = π©′ Bearing Capacity of Shallow Foundation Figure 22 Effective Area (Case V) Bearing Capacity of Shallow Foundation Figure 23 Variation of π¨′ πΉπ πππ π©′ πΉ with ππΉ πΉ for Circular Foundation Bearing Capacity of Shallow Foundation Table 05 Variation of π¨′ πΉπ and π©′ πΉ with ππΉ πΉ for Circular Foundation ππΉ πΉ π¨′ πΉπ π©′ πΉ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.80 2.40 2.00 1.61 1.23 0.93 0.62 0.35 0.12 0.00 1.85 1.32 1.20 0.80 0.67 0.50 0.37 0.23 0.12 0.00
