Ultimate Bearing Capacity

Transcription

1 CE-632 Foundation Analysis and Design Ultimate earing Capacity The load per unit area of the foundation at which shear failure in soil occurs is called the ultimate t bearing capacity. 1

2 Principal Modes of Failure: General Shear Failure: Load / Area ment Settle u Sudden or catastrophic failure Well defined failure surface ulging on the ground surface adjacent to foundation Common failure mode in dense sand 2

3 Principal Modes of Failure: Local Shear Failure: Load / Area u1 Set ttlement u Common in sand or clay with medium compaction Significant settlement upon loading Failure surface first develops right below the foundation and then slowly extends outwards with load increments Foundation movement shows sudden jerks first (at u1 ) and then after a considerable amount of movement the slip surface may reach hthe ground. A small amount of bulging may occur next to the foundation. 3

4 Principal Modes of Failure: Punching Failure: Load / Area u1 Set ttlement u Common in fairly loose sand or soft clay Failure surface does not extends beyond the zone right beneath the foundation Extensive settlement with a wedge shaped soil zone in elastic euilibrium beneath the foundation. Vertical shear occurs around the edges of foundation. After reaching failure load-settlement t curve continues at some slope and mostly linearly. 4

5 Principal Modes of Failure: Rela ative dep pth of fou undation n, D f /* Relative density of sand, D r General Local shear shear 5 10 Punching shear Vesic (1973) Circular Foundation Long Rectangular Foundation * 2L = + L 5

6 Terzaghi s earing Capacity Theory j neglected Rough Foundation Surface u Strip Footing Effective overburden k g Shear Planes 45 φ /2 D f III e a II φ I φ d b II = γ.d f 45 φ /2 i III c - φ soil f Assumption L/ ratio is large plain strain problem D f Shear resistance of soil for D f depth is neglected General shear failure Shear strength is governed by Mohr-Coulomb Criterion 6

7 Terzaghi s earing Capacity Theory 1 2 u. = 2. Pp + 2. Ca.sinφ γ tanφ 4 a u φ φ I b 1 2 u. = 2. P c..sinφ γ tanφ p + 4 C a = /2 P = P + P + γ P cosφ C a.tanφ p p pc p P pγ = due to only self weight of soil in shear zone φ φ d φ φ in shear zone P p P p P pc = due to soil cohesion only (soil is weightless) P p = due to surcharge only 7

8 Terzaghi s earing Capacity Theory Weight term Cohesion term 1 P ( P c ) P 4 2 u. = 2. pγ γ tanφ + 2. pc +..sinφ + 2. p ( γ N γ ) cn c.. N Surcharge term u = cn. c + N γ N. γ Terzaghi s bearing capacity euation Terzaghi s bearing capacity factors 1 K P γ φ 2 1 N tan 1 γ = N 2 cos φ = 2 φ 2cos π φ in rad. N c = ( N 1cot ) φ a = tanφ 4 2 2a e 8

9 Foundation Analysis and Design: Dr. Amit Prashant 9

10 Terzaghi s earing Capacity Theory Local Shear Failure: Modify the strength parameters such as: 2 c m = c 3 2 u = c. N c + N γ N. γ 3 φ m = tan tanφ Suare and circular footing: = 1.3 c. N + N γ N. γ For suare u c = 1.3 c. N + N γ N. For circular γ u c 10

11 Terzaghi s earing Capacity Theory Effect of water table: Case I: D w D f Surcharge, = γ. Dw + γ ( Df Dw) D w Case II: D f D w (D f + ) Surcharge, = γ D. F In bearing capacity euation replace γ by- Dw D f γ = γ + γ γ Case III: D w > (D f + ) ( ) No influence of water table. Another recommendation for Case II: d γ γ = + + H H ( H d ) γ ( H d ) 2 2 w w2 sat 2 w D f Rupture depth: Limit it of influence d = D D w w f ( ) H = 0.5tan 45 + φ 2 11

12 Skempton s earing Capacity Analysis for cohesive Soils ~ For saturated cohesive soil, φ = 0 N = 1, and N γ = 0 Df For strip footing: Nc = with limit of Nc 7.5 For suare/circular footing: N c Df = with limit of Nc 9.0 Df For rectangular footing: Nc = for Df 2.5 L Nc = for Df > 2.5 L = cn. + u Net ultimate bearing capacity,. D c = γ = cn. nu u f u c 12

13 Effective Area Method for Eccentric Loading D f In case of Moment loading e = x M F V y =-2e y A F = L e y = M x F V e x e y L =L-2e y In case of Horizontal Force at some height but the column is centered on the foundation M = F. d y Hx FH M = F. d x Hy FH 13

14 General earing Capacity Euation: (Meyerhof, 1963) = cn.. s. d. i + N.. s. d. i γ. N.. s. d. i u c c c c γ γ γ γ Shape Depth inclination factor factor factor Empirical correction factors = ( ) ( ) ( ) 2 φ.tan N tan 45. e π φ = + Nc N 1 cotφ Nγ = N 1 tan 1.4φ 2 [y Hansen(1970): N γ = 1.5 N 1 tan φ φ [y Vesic(1973): ( ) ( ) ( ) ( ) Nγ = 2 N + 1 tan φ = cn.. s. d. i. g. b + N.. s. d. i. g. b γ. N.. s. d. i. g. b u c c c c c c γ γ γ γ γ γ Ground factor ase factor 14

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16 Meyerhof s Correction Factors: Shape Factors s c for φ φ 10 o 2 φ φ = tan φ L 2 s = sγ = tan 45 + L 2 for lower φ φ value s = = 1 s γ Depth Df φ for φ 10 o Factors dc = tan 45+ Df φ L 2 d = dγ = tan 45+ L 2 for lower φ value d = = 1 d γ Inclination o β Factors ic = i = i γ β = 1 φ 2 16

17 Hansen s Correction Factors: F ( 1 F ) 1/2 Inclination Factors Depth Factors H ic = 1 for φ = 0 2 L. c 05F 0.5F H i = 1 FV + Lc..cotφ For φ = 0 For φ > 0 Df dc = 0.4 for Df D 1 f dc = 0.4 tan for Df > 5 i i c γ 1 1 H = + for φ > 0 2 L. su 5 07F 0.7F H = 1 FV + Lc..cotφ Df dc = for Df D 1 f dc = tan for Df > For Df < For Df > 1 2 tan.( 1 sin ) 2 Df d = + φ φ ( ) 2 D 1 f d = 1+ 2 tan φ. 1 sinφ tan Shape Factors s c = 0.2 ic. for φ = 0 L s 1 i. L sin φ sc = 0.2( 1 2 ic). for φ > 0 L s = i. L = + ( ) ( ) γ γ d γ = 1 Hansen s Recommendation for cohesive saturated soil, φ'=0..( 1 ) = cn + s + d + i + u c c c c

18 Notes: 1. Notice use of effective base dimensions, L by Hansen but not by Vesic. 2. The values are consistent with a vertical load or a vertical load accompanied by a horizontal load H. 3. With a vertical load and a load H L (and either H =0 or H >0) you may have to compute two sets of shape and depth factors s i,, s i,l and d i,, d i,l. For i, L subscripts use ratio L / or D/L. 4. Compute u independently by using (s i, d i ) and (s il, d il ) and use min value for design. 18

19 Notes: 1. Use H i as either H or H L, or both if H L >0. 2. Hansen (1970) did not give an i c for φ>0. The value given here is from Hansen (1961) and also used by Vesic. 3. Variable c a = base adhesion, on the order of 0.6 to 1.0 x base cohesion. 4. Refer to sketch on next slide for identification of angles η and β, footing depth D, location of H i (parallel and at top of base slab; usually also produces eccentricity). Especially notice V = force normal to base and is not the resultant R from combining V and H i.. 19

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21 Note: 1. When φ=0 (and β 0) use N γ = -2sin(±β) in N γ term. γ ( β) γ 2. Compute m = m when H i = H (H parallel to ) and m = m L when H i = H L (H parallel to L). If you have both H and H L use m = (m 2 + m L2 ) 1/2. Note use of and L, not, L. 3. H i term 1.0 for computing i, i γ (always). 21

22 Suitability of Methods 22

23 IS: Recommendations = cn.. s. d. i +. N 1. s. d. i γ. N.. s. d. i Net Ultimate earing capacity: ( ) For cohesive soils nu c c c c = c. N. s. d. i where, N = nu u c c c c N, N, N γ as per Vesic(1973) recommendations c c γ γ γ γ Shape Factors Depth Factors For rectangle, s For suare and circle, d d d c c = s L = + s L = γ L s c = s = s = 0.8 for suare, s = 0.6 for circle Df φ = tan 45+ L 2 D f φ φ = dγ = tan 45+ L 2 = = 1 for φ < 10 o d γ γ for γ φ 10 o Inclination Factors The same as Meyerhof (1963) 23

24 earing Capacity Correlations with SPT-value aue Peck, Hansen, and Thornburn (1974) & IS: Recommendation 24

25 earing Capacity Correlations with SPT-value Teng (1962): For Strip Footing: For Suare and Circular Footing: = 3 N. R. + 5 ( N ). D. R 6 nu w f w = N. R. + 3( N ). D. R 3 nu w f w For D f >, take D f = Water Table Corrections: D w D w R w = R 1 [ w D f Dw D f R = w [ Rw 1 D f D f Limit of influence 25

26 earing Capacity Correlations with CPT-value IS: Recommendation: Cohesionless Soil 1.5 to 2.0 c value is taken as average for this zone nu c D 0.5 f 1 = Schmertmann (1975): c kg N N γ in 0.8 cm 2 (cm) 26

27 earing Capacity Correlations with CPT-value IS: Recommendation: Cohesive Soil = c. N. s. d. i nu u c c c c Soil Type Normally consolidated clays Point Resistance Values Range of Undrained ( c ) kgf/cm 2 Cohesion (kgf/cm 2 ) c < 20 c /18 to c /15 Over consolidated clays c > 20 c /26 to c /22 27

28 earing Capacity of Footing on Layered Soil φ Depth of rupture zone = tan or approximately taken as Case I: Layer-1 is weaker than Layer-2 Design using parameters of Layer -1 Case II: Layer-1 is stronger than Layer-2 Distribute the stresses to Layer-2 by 2:1 method 1 and check the bearing capacity at this level for limit state. 2 Layer-1 Also check the bearing capacity for original Layer-1 Layer-2 foundation level using parameters of Choose minimum value for design φ φ Another approximate method for c -φ soil: For effective depth tan Find average c and φ and use them for ultimate bearing capacity calculation c av ch 1 1+ ch ch = H + H + H tanφ av = tanφ H + tanφ H + tan φ H H + H + H

29 earing Capacity of Stratified Cohesive Soil IS: Recommendation: 29

30 earing Capacity of Footing on Layered Soil: Stronger Soil Underlying Weaker Soil Depth H is relatively small Punching shear failure in top layer General shear failure in bottom layer Depth H is relatively large Full failure surface develops in top layer itself 30

31 earing Capacity of Footing on Layered Soil: Stronger Soil Underlying Weaker e Soil 31

32 earing Capacity of Footing on Layered Soil: Stronger Soil Underlying Weaker Soil earing capacities of continuous footing of with under vertical load on the surface of homogeneous thick bed of upper and lower soil 32

33 earing Capacity of Footing on Layered Soil: Stronger Soil Underlying Weaker Soil For Strip Footing: 2cH 2D K tanφ H H H a 2 f s 1 u = b + + γ1 1+ γ1 t Where, t is the bearing capacity for foundation considering only the top layer to infinite depth For Rectangular Footing: Special Cases: 2 ch 2 2 tan a Df Ks φ1 u = b γ γ1 t H H L L H 1. Top layer is strong sand and bottom layer is saturated soft clay c = 1 0 φ = Top layer is strong sand and bottom layer is weaker sand c = c 2 = Top layer is strong saturated clay and bottom layer is weaker saturated clay φ 1 = 0 φ 2 = 0 33

34 Eccentrically Loaded Foundations Q M M e = Q Q = + L 6M L max 2 max Q = 1+ L 6e Q = L 6M L min 2 min Q = 1 L 6e e 1 e For > 6 There will be separation of foundation from the soil beneath and stresses will be redistributed. = 2e Use for s, and, L for dc, d, d γ to obtain L c, s, s = L γ u Q. u = u A The effective area method for two way eccentricity becomes a little more complex than what is suggested above. It is discussed in the subseuent slides 34

35 Determination of Effective Dimensions for Eccentrically Loaded d foundations (Highter and Anders, 1985) Case I: el 1 e 1 and L e 1 = 2 L e e L L 1 L 1 3 = L 2 3 L e L A 1 L 2 = L = max (, L ) A = L 35

36 Determination of Effective Dimensions for Eccentrically Loaded foundations (Highter and Anders, 1985) Case II: el L e < 0.5 and 0 < < 1 6 L2 e e L L 1 L 1 A ( L L ) 2 L = max, L = 1+ 2 A ( ) 1 1 = L 36

37 Determination of Effective Dimensions for Eccentrically Loaded foundations (Highter and Anders, 1985) Case III: el 1 e and L < 6 < < 1 e e L L 1 A = L + L = L ( ) 2 A = L A 37

38 Determination of Effective Dimensions for Eccentrically Loaded foundations (Highter and Anders, 1985) Case IV: el L 1 1 e 1 < and < 6 6 e L e L 2 A 1 = L + + L + 2 L A L = L = L ( )( )

39 Determination of Effective Dimensions for Eccentrically Loaded foundations (Highter and Anders, 1985) Case V: Circular foundation e R R A L = 39

40 Meyerhof s (1953) area correction based on empirical correlations: (American Petroleum Institute, t 1987) 40

41 earing Capacity of Footings on Slopes Meyerhof s (1957) Solution = cn γγ N γ u c Granular Soil c = 0 u = 0.5γ N γ 41

42 earing Capacity of Footings on Slopes Meyerhof s (1957) Solution Cohesive Soil φ = 0 u = cn c N s γ H = c 42

43 earing Capacity of Footings on Slopes Graham et al. (1988), ased on method of characteristics For D f =

44 earing Capacity of Footings on Slopes Graham et al. (1988), ased on method of characteristics 1000 For 100 D f =

45 earing Capacity of Footings on Slopes Graham et al. (1988), ased on method of characteristics ti For D f =

46 earing Capacity of Footings on Slopes Graham et al. (1988), ased on method of characteristics ti For D f 1.0 = 46

47 earing Capacity of Footings on Slopes owles (1997): A simplified approach f g u α = 45+φ /2 f' g' u e D f 45 φ /2 d a α b α c e' 45 φ /2 d' r a' α r o b' α c' e' 45 φ /2 f' d' g' a' u α b' α c' Compute the reduced factor N c as: L N = N L c abde c. L abde Compute the reduced factor N as: Aaefg N = N. A aefg 47

48 Soil Compressibility Effects on earing Capacity Vesic s (1973) Approach Use of soil compressibility factors in general bearing capacity euation. These correction factors are function of the rigidity of soil Rigidity Index of Soil, I r : Critical Rigidity Index of Soil, I cr : I r Irc Gs = c + σ tanφ = 0.5. e vo L φ tan 45 2 Compressibility Correction Factors, c c, c g, and c σ vo = γ. ( D f + /2 ) For Ir I c 1 rc c = c = c γ = /2 For I r < I rc 3.07.sin φ.log tan φ + L 1+ sinφ 10 ( I r ) c = c = e 1 γ For φ = 0 c = c log I L + 1 c For φ > 0 cc = c N tanφ r 48