Ultimate Bearing Capacity

Transcription

1 CE-63 Foundation Analysis and Design Ultimate earing Capacity The load per unit area o the oundation at which shear ailure in soil occurs is called the ultimate bearing capacity. Principal Modes o Failure: General Shear Failure: oad / Area ment u Settle Sudden or catastrophic ailure Well deined ailure surace ulging on the ground surace adjacent to oundation Common ailure mode in dense sand Principal Modes o Failure: ocal Shear Failure: oad / Area u ttlement Set u Common in sand or clay with medium compaction Signiicant settlement upon loading Failure surace irst develops right below the oundation and then slowly extends outwards with load increments Foundation movement shows sudden jerks irst (at u ) and then ater a considerable amount o movement the slip surace may reach the ground. A small amount o bulging may occur next to the oundation. 3

2 Principal Modes o Failure: Punching Failure: oad / Area u ttlement Set u Common in airly loose sand or sot clay Failure surace does not extends beyond the zone right beneath the oundation Extensive settlement with a wedge shaped soil zone in elastic euilibrium beneath the oundation. Vertical shear occurs around the edges o oundation. Ater reaching ailure load-settlement curve continues at some slope and mostly linearly. 4 Principal Modes o Failure: Relative depth o ou undation, D /* Relative density o sand, D r Punching shear ocal shear General shear Vesic (973) Circular Foundation ong Rectangular Foundation * = + 5 Terzaghi s earing Capacity Theory j neglected g Shear Planes Rough Foundation Surace 45 φ / D III e a II u φ I φ d b II Strip Footing Eective overburden = γ.d 45 φ / i III c - φ soil k Assumption / ratio is large plain strain problem D Shear resistance o soil or D depth is neglected General shear ailure Shear strength is governed by Mohr-Coulomb Criterion 6

3 Terzaghi s earing Capacity Theory u. =.Pp +.Ca.sin φ γ tan φ 4 u b a φ Ca= / cosφ φ Pp u. =.Pp +.c.sin φ γ tan φ 4 φ I Pp = Ppγ + Ppc + Pp Ca.tanφ Ppγ = due to only sel weight o soil in shear zone φ d Pp Ppc = due to soil cohesion only (soil is weightless) Pp = due to surcharge only 7 Terzaghi s earing Capacity Theory Weight term Cohesion term u. =.Ppγ γ tan φ + (.Ppc +.c.sin φ ) +.Pp 4. ( 0.5γ.Nγ ) Surcharge term.c.nc..n Terzaghi s bearing capacity euation u = c.n c +.N + 0.5γ.Nγ Terzaghi s bearing capacity actors Nγ = K tan φ Pγ φ cos N c = ( N ) cot φ e a φ cos π φ in rad. a= tan φ 4 N = 8 9 3

4 Terzaghi s earing Capacity Theory ocal Shear Failure: Modiy the strength parameters such as: c m = c 3 u = c. N c + N γ N. γ 3 φ m = tan tanφ 3 Suare and circular ooting: =.3 c. N + N γ N. γ u c =.3 c. N + N γ N. γ u c For suare For circular 0 Terzaghi s earing Capacity Theory Eect o water table: Case I: D w D Surcharge, = γ. Dw + γ ( D Dw) D w Case II: D D w (D + ) Surcharge, = γ. DF In bearing capacity euation replace γ by- Dw D γ = γ + ( γ γ ) Case III: D w > (D + ) No inluence o water table. Another recommendation or Case II: d γ = ( H + dw) γsat γ + ( H d ) w w H H D Rupture depth: imit o inluence dw = Dw D H = 0.5tan ( 45 + φ ) Skempton s earing Capacity Analysis or cohesive Soils ~ For saturated cohesive soil, φ = 0 N =, and N γ = 0 For strip ooting: D Nc = with limit o Nc 7.5 D For suare/circular Nc = with limit o Nc 9.0 ooting: For rectangular ooting: D Nc = or D.5 Nc = or D >.5 = cn. + u c Net ultimate bearing capacity, = u γ. D u =. nu cn c 4

5 Eective Area Method or Eccentric oading D In case o Moment loading M y ex = F V =-e y A F = e y M x = F V e x e y =-e y In case o Horizontal Force at some height but the column is centered on the oundation M = F. d y Hx FH M = F. d x Hy FH 3 General earing Capacity Euation: (Meyerho, 963) = cn.. s. d. i + N.. s. d. i γ. N.. s. d. i u c c c c γ γ γ γ Shape actor Depth actor inclination actor Empirical correction actors ( ) φ.tan N tan 45. e π φ = + Nc = N cotφ Nγ = ( N ) tan (.4φ ) [y Hansen(970): Nγ =.5 N tan φ [y Vesic(973): ( ) ( ) ( ) ( ) Nγ = N + 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 actor ase actor 4 5 5

6 Meyerho s Correction Factors: Shape Factors φ sc = + 0. tan 45+ or φ 0 o s sγ 0. tan 45 φ = = + + or lower φ value s = = s γ Depth D φ or φ 0 o Factors dc = + 0. tan 45+ D φ d = dγ = + 0. tan 45+ or lower φ value d = = d γ Inclination Factors o β ic = i = 90 i γ β = φ 6 Hansen s Correction Factors: FH ic = or φ = 0. c 5 0.5F H i = FV + c..cotφ Inclination Factors Depth Factors Shape Factors For φ = 0 D dc = 0.4 or D D dc = 0.4 tan or D > ( F ) / H ic = + or φ > 0. su 5 0.7F H iγ = FV + c..cotφ For φ > 0 D dc = or D D dc = tan or D > For D < For D > tan.( sin ) D d = + φ φ ( ) D d = + tan φ. sinφ tan sc = 0. ic. or φ = 0 s = + i. ( ) sinφ d γ = sc = 0.( ic). or φ > 0 sγ = 0.4 iγ.( ) Hansen s Recommendation or cohesive saturated soil, φ'=0..( ) = cn + s + d + i + u c c c c Notes:. Notice use o eective base dimensions, by Hansen but not by Vesic.. 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 (and either H =0 or H >0) you may have to compute two sets o shape and depth actors s i,, s i, and d i,, d i,. For i, subscripts use ratio / or D/. 4. Compute u independently by using (s i, d i ) and (s i, d i ) and use min value or design. 8 6

7 Notes:. Use H i as either H or H, or both i H >0.. Hansen (970) did not give an i c or φ>0. The value given here is rom Hansen (96) and also used by Vesic. 3. Variable c a = base adhesion, on the order o 0.6 to.0 x base cohesion. 4. Reer to sketch on next slide or identiication o angles η and β, ooting depth D, location o H i (parallel and at top o base slab; usually also produces eccentricity). Especially notice V = orce normal to base and is not the resultant R rom combining V and H i Note:. When φ=0 (and β 0) use N γ = -sin(±β) in N γ term.. Compute m = m when H i = H (H parallel to ) and m = m when H i = H (H parallel to ). I you have both H and H use m = (m + m ) /. Note use o and, not,. 3. H i term.0 or computing i, i γ (always). 7

8 Suitability o Methods IS: Recommendations = cn.. s. d. i +. N. 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 = 5.4 nu u c c c c N, N, N γ as per Vesic(973) recommendations c c γ γ γ γ Shape Factors Depth Factors Inclination Factors sc = + 0. s 0. = + s 0.4 For rectangle, = γ For suare and circle, s c = 3.3 s =. sγ = 0.8 or suare, sγ = 0.6 or circle D φ dc = + 0. tan 45+ D φ d = dγ = + 0. tan 45+ or φ 0 o = = or φ < 0 o d d γ The same as Meyerho (963) 3 earing Capacity Correlations with SPT-value Peck, Hansen, and Thornburn (974) & IS: Recommendation 4 8

9 earing Capacity Correlations with SPT-value Teng (96): For Strip Footing: For Suare and Circular Footing: nu = w w ( ) N R N D R 6 nu = N. R. w + 3( 00 + N ). D. R w 3 For D >, take D = Water Table Corrections: D w D w Rw = [ Rw D Dw D R w = [ R w D D imit o inluence 5 earing Capacity Correlations with CPT-value IS: Recommendation: Cohesionless Soil.5 to.0 c value is taken as average or this zone nu c Schmertmann (975): c kg N N γ in 0.8 cm D 0.5 = (cm) 0 6 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 o Undrained ( c ) kg/cm Cohesion (kg/cm ) c < 0 c /8 to c /5 Over consolidated clays c > 0 c /6 to c / 7 9

10 earing Capacity o Footing on ayered Soil φ Depth o rupture zone = tan 45 + or approximately taken as Case I: ayer- is weaker than ayer- Design using parameters o ayer - Case II: ayer- is stronger than ayer- Distribute the stresses to ayer- by : method and check the bearing capacity at this level or limit state. ayer- Also check the bearing capacity or original ayer- oundation level using parameters o ayer- Choose minimum value or design φ Another approximate method or c -φ soil: For eective depth tan 45 + Find average c and φ and use them or ultimate bearing capacity calculation ch + ch + ch tanφh+ tanφh + tan φ3h cav = tanφav = H + H + H +... H + H + H earing Capacity o Stratiied Cohesive Soil IS: Recommendation: 9 earing Capacity o Footing on ayered Soil: Stronger Soil Underlying Weaker Soil Depth H is relatively small Punching shear ailure in top layer General shear ailure in bottom layer Depth H is relatively large Full ailure surace develops in top layer itsel 30 0

11 earing Capacity o Footing on ayered Soil: Stronger Soil Underlying Weaker Soil 3 earing Capacity o Footing on ayered Soil: Stronger Soil Underlying Weaker Soil earing capacities o continuous ooting o with under vertical load on the surace o homogeneous thick bed o upper and lower soil 3 earing Capacity o Footing on ayered Soil: Stronger Soil Underlying Weaker Soil For Strip Footing: ch D a Ks tanφ u = b + + γh + γh t H Where, t is the bearing capacity or oundation considering only the top layer to ininite depth For Rectangular Footing: ch tan a D Ks φ u = b + + γh γh t H Special Cases:. Top layer is strong sand and bottom layer is saturated sot clay c = 0 φ = 0. Top layer is strong sand and bottom layer is weaker sand c = 0 c = 0. Top layer is strong saturated clay and bottom layer is weaker saturated clay φ = 0 φ = 0 33

12 Eccentrically oaded Foundations Q M = Q 6M + max min M e = Q = Q 6M max min Q 6e = + Q 6e = e For e > There will be separation 6 o oundation rom the soil beneath and stresses will be redistributed. = e Use or s, and, or dc, d, d γ to obtain c, s, s = γ u Q. u = u A The eective area method or two way eccentricity becomes a little more complex than what is suggested above. It is discussed in the subseuent slides 34 Determination o Eective Dimensions or Eccentrically oaded oundations (Highter and Anders, 985) Case I: e e and e = e e 3 3e = A = ( ) = max, A = 35 Determination o Eective Dimensions or Eccentrically oaded oundations (Highter and Anders, 985) Case II: e e < 0.5 and 0 < < 6 e e A ( ) = max, = + A ( ) = 36

13 Determination o Eective Dimensions or Eccentrically oaded oundations (Highter and Anders, 985) Case III: e e and 0 < < < e e A = + = ( ) A = 37 Determination o Eective Dimensions or Eccentrically oaded oundations (Highter and Anders, 985) e Case IV: e < and < 6 6 e e A = A = = ( )( ) 38 Determination o Eective Dimensions or Eccentrically oaded oundations (Highter and Anders, 985) Case V: Circular oundation er R A = 39 3

14 Meyerho s (953) area correction based on empirical correlations: (American Petroleum Institute, 987) 40 earing Capacity o Footings on Slopes Meyerho s (957) Solution u = cn c + 0.5γ N γ Granular Soil c = 0 u = 0.5γ N γ 4 earing Capacity o Footings on Slopes Meyerho s (957) Solution Cohesive Soil φ = 0 u = cn c γ H Ns = c 4 4

15 earing Capacity o Footings on Slopes Graham et al. (988), ased on method o characteristics 000 For 00 D 0 = earing Capacity o Footings on Slopes Graham et al. (988), ased on method o characteristics 000 For 00 D 0 = earing Capacity o Footings on Slopes Graham et al. (988), ased on method o characteristics For D 0.5 = 45 5

16 earing Capacity o Footings on Slopes Graham et al. (988), ased on method o characteristics For D.0 = 46 earing Capacity o Footings on Slopes owles (997): A simpliied approach g u α = 45+φ / ' g' u e D 45 φ / d a α b α c e' 45 φ / d' r a' α α r o b' c' e' 45 φ / ' d' g' u a' α α b' c' Compute the reduced actor N c as: abde N c = Nc. abde Compute the reduced actor N as: Aaeg N = N. A aeg 47 Soil Compressibility Eects on earing Capacity Vesic s (973) Approach Use o soil compressibility actors in general bearing capacity euation. These correction actors are unction o the rigidity o soil G Rigidity Index o Soil, I s r : Ir = c + σ vo tanφ Critical Rigidity Index o Soil, I cr : φ tan 45 Irc = 0.5. e / Compressibility Correction Factors, c c, c g, and c For Ir I rc cc = c = c γ = For I r < I rc 3.07.sin φ.log0 (. I ) tan r φ + + sinφ c = cγ = e For φ = 0 cc = log Ir c For φ > 0 cc = c N tanφ ( D ) σ = γ. + / vo 48 6