Bearing Capacity Equation



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CHAPTER 7 BEARING CAPACITY OF SOILS AND SETTLEMENT OF SHALLOW FOUNDATIONS COLLAPSE LOAD FROM LIMIT EQUILIBRIUM Moment equilibriumequation for a circular failure mechanism Pu × B − s u πB × B = 0 2 Collapse load Pu = 6.28Bs u Moment equilibrium equation about point O P u (R cos θ B ) − su [(π - 2θ)R]R = 0 2 Collapse load found from the moment about point O Pu = su [(π - 2θ)R ]R su (π - 2θ)R = B B ) (cos θ (R cos θ - ) 2R 2 Partial derivatives of collapse load equation with respect to R and θ ∂ Pu 4su R (π - 2θ) . (R cos θ - B) = =0 2 ∂R (2R cos θ - B ) ∂ Pu 4su R 2 (B - 2R cos θ + πR sin θ - 2Rθ sin θ) =0 = ∂θ (2R cos θ - B )2 Collapse load with R and θ values obtained from partial derivative equations Pu = 5.52Bs u More exact collapse load solution using complex analysis Pu = 5.14Bs u BEARING CAPACITY EQUATIONS Terzaghi’s Bearing Capacity Equations Ultimate net bearing capacity equations TSA (Total Stress Analysis): q ult = 5.14s u s c + γD f ESA (Effective Stress Analysis): q ult = γD f N q s q + 0.5γBN γ s γ 14s u s c d c + γD f ESA: q ult = γD f N q s q d q + 0.14s u d c i c + γD f ESA: q ult = γD f N q d q i q + 0.1 K p φ′ 1 + sin φ′ )= 2 1 − sin φ′ Df B where K p = tan 2 (45 o + Load inclination factor for loads inclined at an angle θ to the vertical in the direction of footing width .4φ′) Shape and depth factors s c = 1 + 0 .2 ) + γD f .2 B B s q = s γ = 1 + 0.2 B B B s q = 1 + tan φ′ s γ = 1 − 0. Nq is the same as Terzaghi.5γBN γ s γ d γ Bearing capacity equation for inclined loads TSA: q ult = 5. f ≤ 2.5 B L B Meyerhof’s Bearing Capacity Equation Bearing capacity equation for vertical loads TSA: q ult = 5.Bearing Capacity Factors N q = e π tan φ′ tan 2 (45 o + N γ = 2( N q + 1) tan φ′ φ′ ) 2 Shape Factors s c = 1 + 0 .2 Df B d q = d γ = 1 + 0.4 L L L Skempton’s Bearing Capacity Equation Bearing capacity equation based on a TSA for rectangular and square footings on clay q ult = 5s u (1 + 0. 2 N γ = ( N q − 1) tan(1.5γBN γ d γ i γ Meyerhof’s bearing capacity factors N q = e π tan φ′ tan 2 (45 o + φ′ ) .2 Df D B )(1 + 0.1K p L L d c = 1 + 0. ⎛ θo ⎞ i c = i q = ⎜1 − ⎟ ⎜ 90 ⎟ ⎠ ⎝ 2 ⎛ θo ⎞ i γ = ⎜1 − ⎟ ⎜ φ′ ⎟ ⎠ ⎝ 2 Load inclination factor for loads inclined at an angle θ to the vertical in the direction of the footing length for a surface footing (Df = 0) ⎡ ⎛ ⎤ α ⎞ ⎛ sin θ ⎞ i c = cos θ⎢1 − ⎜1 − a ⎟ sin θ⎥ i q = i γ = cos θ⎜1 − ⎟ ⎜ sin φ′ ⎟ ⎠ ⎝ ⎣ ⎝ π+2⎠ ⎦ ALLOWABLE BEARING CAPACITY AND FACTOR OF SAFETY Allowable bearing capacity qa = q ult FS Factor of safety FS = q ult σd ECCENTRIC LOADS Modified width and length for a rectangular footing B′ = B − 2e B and L′ = L − 2e L Eccentricities in the directions of width and length eB = My P .eL = Mx P Stresses due to a vertical load at an eccentricity σ= P My P Pey P Pe ± = ± = ± A I A I A Z Maximum and minimum vertical stresses along the X-axis σ max = P Pe P ⎛ 6e B ⎞ P Pe P ⎛ 6e B ⎞ + = = ⎜1 + ⎟ . σ min = − ⎜1 − ⎟ A Z BL ⎝ L ⎠ A Z BL ⎝ L ⎠ Ultimate load . σ min = − ⎜1 − ⎟ A Z BL ⎝ B ⎠ A Z BL ⎝ B ⎠ Maximum and minimum vertical stresses along the Y-axis σ max = P Pe P ⎛ 6e L ⎞ P Pe P ⎛ 6e L ⎞ + = = ⎜1 + ⎟ . 38 ( ) Shape. embedment (trench).16⎜ w ⎜A ⎝ b Elastic settlement modified to account for embedment ρe = q s B′(1 − ν 2 ) u I s µ′ emb Eu where µ ′ = 1 − 0.04 D f ⎧ 4 ⎛ A b ⎞⎫ ⎟⎬ ⎨1 + ⎜ B ⎩ 3 ⎝ 4L2 ⎠⎭ ⎞ ⎟ ⎟ ⎠ 0.Pu = q ult B′L′ SETTLEMENT Maximum tolerable settlement ⎛δ⎞ ρ max = R ⎜ ⎟ ⎝ ⎠ max SETTLEMENT CALCULATIONS Immediate Settlement Elastic settlement ρe = P 1 − ν 2 µ s µ emb µ wall u EuL −0.08 emb Df 4 B′ ) (1 + 3 L′ B′ Primary Consolidation Settlement Method to modify the one-dimensional consolidation equation to account for lateral stresses but not lateral strains (ρ ) pc SB = ∫ m v ∆u o dz O H Excess pore water pressure .45⎜ b2 ⎟ ⎝ 4L ⎠ µ emb = 1 − 0. and side wall factors ⎛A ⎞ µ s = 0.54 µ wall ⎛A = 1 − 0. if q ap > σ′ zc c 3 ⎠ ⎝ . 1974) ⎝ z ⎠ Groundwater correction factor cW = 1 1 z + 2 2 (D f + B) Corrected N value Ncor = CN CW N Allowable bearing capacity for a shallow footing q a = 0.⎡ ⎤ ∆σ 3 (1 − A )⎥ ∆u = ∆σ1 ⎢A + ∆σ1 ⎣ ⎦ One-dimensional primary consolidation settlement (ρ ) pc SB H ⎡ ⎤ ∆σ 3 (1 − A )⎥dz = ∑ (m v ∆σ z H)µ SB = ∑ ρ pc µ SB = ∫ m v ∆σ1 ⎢A + O ∆σ1 ⎣ ⎦ DETERMINATION OF BEARING CAPACITY AND SETTLEMENT OF COARSE-GRAINED SOILS FROM FIELD TESTS Bearing Capacity and Settlement from the Standard Penetration Test (SPT) Correction factor for overburden pressures ⎛ 95..77 log10 ⎜ ⎟ ⎜ σ′ ⎟ .4 Settlement if the sand is overconsolidated 2 ⎞ ⎛ ρ = f 1f s ⎜ q ap − σ′ ⎟B 0. c N ≤ 2 (Liao and Whitman.7 I c .8 ⎞ 2 cN = ⎜ ⎜ σ′ ⎟ . al. c N ≤ 2 .71 N 1 . σ′z > 24kPa (Peck et.41N cor ρ a (kPa) Settlement of a footing in a normally consolidated sand at the end of construction ρ = f s f1q ap B0.7 I c Compressibility index Ic = compressibility index = 1. 1985) ⎟ ⎝ z ⎠ 1 ⎛ 1916 ⎞ c N = 0. 7 Ic .5 σ′z t ≥ 0.2 + ( I cp − 0.ρ = qB0.1) z z 1 for ≤ B B 2 2⎛ z 1⎞ z 1 I co = 1 − ⎜ − ⎟ for 2 ≥ > 3⎝ B 2⎠ B 2 Influence factor for plane strain conditions: L > 10B I co = 0.1 Influence factor for axisymmetric conditions: L=B I co = 0.2 ) z z for ≤1 B B ⎡ 1 ⎛ z ⎞⎤ z I co = I cp ⎢1 − ⎜ − 1⎟ ⎥ for 4 ≥ > 1 B ⎣ 3 ⎝ B ⎠⎦ Settlement from Plate Load Tests Plate settlement when sand behaves like an elastic material 1 − (ν ′) 2 ρ p = q ap B p Ip E′ Relationship between settlement and plate settlement ⎛ ⎜ 2 ρ = ρp ⎜ Bp ⎜ ⎜1+ B ⎝ ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ 2 .1 + 2 ( I cp − 0.5 and ct = creep factor = 1.0 + A log 10 q net 0 . 1978 ρ= n (I ) cDct q net ∑ co i ∆z i β i =1 (q c )i where cD = depth factor = 1 − 0. if q ap < σ′zc 3 Settlement from Cone Penetration Test Settlement proposed by Schhmertmann et al.
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