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Bearing Capacity of Shallow Foundations General Foundations for structures are generally classified as "__________" and "____________". Shallow is a relative term, and may be generally defined as foundations less than approximately 3 m or less than the breadth of the footing. Deep foundations generally refer to piled foundations, whereas shallow foundations include pad foundations, raft foundations, and strip footings. L L B B Strip Footing Spread Footing The performance and functional viability of a foundation depends on the interaction between the structure which is supported and on the founding material. The behaviour of the soil depends on the __________ and ___________ of the foundation, hence the bearing capacity is not simply a function of the soil, but rather is also a function of the specific foundation arrangement. What we find from a practical perspective, is that _________________ generally limit the capacity of a foundation, rather than the ______________________________ of the soil. Hence the design procedure for foundations must include deformation considerations. EnvE 434 1 01/14/02 Bearing Capacity There are then three main criteria which must be considered: a. Adequate depth. b. Limiting settlement. Typical limiting values for settlement are: SANDS: Max total settlement 40 mm for isolated footings 40 - 65 mm for rafts Max differential settlement between adjacent columns 25 mm CLAYS: Max total settlement 65 mm for isolated footings 65 - 100 mm for rafts Max differential settlement between adjacent columns 40 mm c. Factor of safety against shear failure. EnvE 434 2 01/14/02 Bearing Capacity Modes of Failure There are three principal modes of shear failure: 1. 2. 3. General shear failure results in a clearly defined plastic yield slip surface beneath the footing and spreads out one or both sides, eventually to the ground surface. Failure is sudden and will often be accompanied by severe tilting. Generally associated with _____ _______________________________________________________________________. Local shear failure results in considerable vertical displacement prior to the development of noticeable shear planes. These shear planes do not generally extend to the soil surface, but some adjacent bulging may be observed, but little tilting of the structure results. Generally occurs in ____________________________ soils. EnvE 434 3 01/14/02 Bearing Capacity Punching shear failure occurs in __________________ soils, and there is little or no development of planes of shear failure in the underlying soil. Slip surfaces are generally restricted to vertical planes adjacent to the footing, and the soil may be dragged down at the surface in this region. Definitions of Bearing Capacity _____________________________, qu, is the maximum bearing pressure at which the supporting ground is expected to fail in shear. P q GWT D hw B There is a stress at the base of the footing due to the backfill and the footing mass This can be approximated as: σ'o = The net ultimate bearing capacity can be defined as: qnf = EnvE 434 4 01/14/02 Bearing Capacity _____________________________, qall is the ultimate bearing capacity divided by an appropriate factor of safety. Generally for bearing capacity of shallow foundations, the factor of safety is __________. It has been found from experience that this is appropriate in order to limit deformations to those allowable. The applied stresses at the base of the footing due to the applied load is defined as: q= The net (increased) applied stress at the footing base is defined as: qnet The Factor of safety for bearing capacity is then defined as F= or in terms of allowable bearing stress, qall = q, thus F= Solving for the allowable bearing stress we obtain qall In the Canadian foundation Engineering Manual (CFEM) this expression is simplified to: qall Which is conservative (less than the equation above by σ'o - σ'o/F, approx 0.67 σ'o) EnvE 434 5 01/14/02 Bearing Capacity _______________________________, qp, is a conservative estimate of bearing capacity for preliminary design purposes, taken from empirical data and local experience. Some typical values are listed below. Type of soil qp (kPa) Remarks Cohesionless soil Providing width B > 1m and compact gravel or sand/gravel >600 groundwater level > B below medium-dense gravel or sand/gravel 200 - 600 base of footing. loose gravel or sand/gravel <200 compact sand >300 medium-dense sand 100 - 300 loose sand <100 Cohesive soil These soils are susceptible to very stiff boulder clays; hard clays 300 - 600 long-term settlement. stiff clays 150 - 300 firm clays 75 - 150 soft clays and silts <75 very soft clays and silts not applicable Ultimate Bearing Capacity of Shallow Foundations As with most geotechnical applications which we've studied, the behaviour is first idealized in the form a simplified model as the basis for a mathematical formulation. This model and formulation are then modified by empirical corrections to provide a good correlation between theoretical and observed behaviour. The first case considered is that of an ________________________________________ of width B on a homogeneous, isotropic, weightless soil. A general shear failure is assumed and the condition of limit equilibrium is analyzed for the soil which behaves as and elastic-perfectly plastic material. EnvE 434 6 01/14/02 Bearing Capacity The failure of the soil is divided into 3 zones as shown below: 1) 2) , and 3) Slip is assumed to occur on the planes as shown. If the footing is installed beneath the soil surface, the overburden pressure on the soil adjacent to the footing is considered as a surcharge on the notional ground surface σo = (where D is the depth from the soil surface to footing base) When failure is reached, the soil wedge beneath the footing (ABF) is displaced downward developing a Rankine active state such that σ1 is _____________ σ3 is _____________ The adjacent radial sections are then forced to rotate sideways and the principal stresses rotate as well. The passive wedges are forced upwards and away from the footing by the radial rotating sections, and for these sections the stresses are: σ1 is _____________ σ3 is _____________ Under undrained conditions the angle α in the figure is equal to: α = 45 ± φ'/2 and φ = 0, thus the angles for the active and passive wedges are both equal to 45 and hence the arc of the radial section is circular. EnvE 434 7 01/14/02 Bearing Capacity For these conditions it has been determined that the bearing capacity may be calculated from: qu = For drained conditions (φ > 0) the angles are as shown below, and as a consequence the curve of the radial portion is not circular but rather is defined as a log spiral. For these conditions it has been determined that the bearing capacity may be calculated from: qu = where Nc and Nq are dimensionless factors dependent on φ'. In order to account for the weight of the soil (so far it has been considered weightless) another term must be added accounting for density. This was done by Terzaghi in 1943 for a strip footing of width B resulting in the equation: qu = The parameters Nc, Nq, and Nγ are referred to as the Bearing Capacity Factors. The values of these parameters can be calculated from the equations below, and are tabulated on the following page. Nq = e π tan φ' tan2 (45 + φ' / 2) Reissner 1924) Nc = cot φ' (Nq - 1) Prandtl (1921) Nγ = 1.8 (Nq - 1) tan φ' Hansen (1961) (Note: CFEM uses 1.5 vs. 1.8) EnvE 434 8 01/14/02 Bearing Capacity Bearing Capacity Factors φ' Nc Nq Nγ φ' Nc Nq Nγ φ' Nc Nq Nγ 0 5.14 1.00 0.00 15 11.0 3.94 1.42 30 30.1 18.4 18.1 1 5.38 1.09 0.00 16 11.6 4.34 1.72 31 32.7 20.6 21.2 2 5.63 1.20 0.01 17 12.3 4.77 2.08 32 35.5 23.2 24.9 3 5.90 1.31 0.03 18 13.1 5.26 2.49 33 38.6 26.1 29.3 4 6.19 1.43 0.05 19 13.9 5.80 2.97 34 42.2 29.4 34.5 5 6.49 1.57 0.09 20 14.8 6.40 3.54 35 46.1 33.3 40.7 6 6.81 1.72 0.14 21 15.8 7.07 4.19 36 50.6 37.8 48.1 7 7.16 1.88 0.19 22 16.9 7.82 4.96 37 55.6 42.9 56.9 8 7.53 2.06 0.27 23 18.0 8.66 5.85 38 61.4 48.9 67.4 9 7.92 2.25 0.36 24 19.3 9.60 6.89 39 67.9 56.0 80.1 10 8.34 2.47 0.47 25 20.7 10.7 8.11 40 75.3 64.2 95.4 11 8.80 2.71 0.60 26 22.3 11.9 9.53 41 83.9 73.9 114 12 9.28 2.97 0.76 27 23.9 13.2 11.2 42 93.7 85.4 137 13 9.81 3.26 0.94 28 25.8 14.7 13.1 43 105 99.0 165 14 10.4 3.59 1.16 29 27.9 16.4 15.4 44 118 115 199 45 134 135 241 Values of Nc after Prandtl (1921) 46 152 159 294 Nq after Reissner (1924) 47 174 187 359 Nγ after Hansen (1961) 48 199 222 442 49 230 265 548 50 267 319 682 Bearing Capacity Factors 50 45 40 35 30 Factor 25 Nc 20 15 10 Nq 5 Nγ 0 0 10 20 30 40 50 φ' EnvE 434 9 01/14/02 Bearing Capacity Modifications for different foundation configurations The above calculations for bearing capacity were based upon an infinitely long strip footing. Modification to the calculation must be made for different shaped footings, and for deeper foundations as the failure mode changes. The bearing capacity equation becomes: qu = Where the values of sc sq and sγ are shape factors, and dc dq and dγ are depth factors calculated based on the values tabulated below: Footing Shape sc sq sγ Strip 1.00 1.00 1.00 Rectangle 1 + (B/L) * (Nq/ Nc) 1 + (B/L) tanφ' 1 - 0.4 B/L Circle or square 1 + (Nq/Nc) 1 + tanφ' 0.6 dc = 1 + 0.2 (Kp)1/2 (D/B) and dq = dγ = 1 + 0.1 (Kp) (D/B) The depth factors are generally quite small except for deeper footings Soft Soils Terzaghi recommended that the bearing capacity calculations be modified in soft cohesive soils or low density cohesionless soils to account for the fact that the mode of failure was different (local or punching vs general). Recommended modifications are: Cohesive soils (modify c): c= tanφ' = Cohesionless soil (modify φ') CF = where Dr = relative density 0 < Dr < 0.67 tanφ' = CF tanφ'measured EnvE 434 10 01/14/02 Bearing Capacity Presence of the Water Table In granular soils, the presence of water in the soil can substantially reduce the bearing capacity. Case 1 Case 2 Case 3 Case 4 GWT q GWT D GWT D+B B GWT Case 1:use γ' for the γDNq and ½BγNγ terms Overburden Beneath Footing Case 2:for the γDNq = σ'Nq term calculate the effective stress at the depth of the footing σ' = σ-u = γD - γwhw, and for the ½BγN use γ'. Case 3:use γ for the γDNq term, and use γ' for the ½BγNγ term. Case 4:use γ for the γDNq and ½BγNγ terms. In cohesive soils for short-term, end-of-construction conditions use: γ = γt and φ = 0 Nc = 5.14, Nq = 1, and Nγ = 0 Thus qu = 5.14c + γt D EnvE 434 11 01/14/02 Bearing Capacity General Observations about Bearing Capacity 1. The cohesion term dominates in cohesive soils. 2. The depth term (γ D Nq) dominates in cohesionless soils. Only a small increase in D increases qu substantially. 3. The base width term (0.5 γ B Nγ) provides some increase in bearing capacity for both cohesive and cohesionless soils. In cases where B < 3 to 4 m this term could be neglected with little error. 4. No one would place a footing on the ground surface of a cohesionless soil mass. 5. It's highly unlikely that one would place a footing on a cohesionless soil with a Dr < 0.5. If the soil is loose, it would be compacted in some manner to a higher density prior to placing footings on it. 6. Where the soil beneath the footing is not homogeneous or is stratified, some judgment must be applied to determining the bearing capacity. In practice, For the short term : we typically use φ = 0 so the bearing capacity equation simplifies to qu = For long-term performance, we usually use SPT blow counts and the charts. BEARING CAPACITY FROM SPT RESULTS (Ref: CFEM) • The allowable bearing pressure, qa, for a footing on sand can be estimated from the results of an SPT test by means of the relationship between the SPT index, N, and the footing width, as given in Fig. 10.1. • Values determined in this manner correspond to the case where the groundwater table is located deep below the footing foundation elevation. • If the water table rises to the foundation level, no more than half the pressure values indicated in Fig 10.1 should be used. EnvE 434 12 01/14/02 Bearing Capacity • The charts are based on SPT indices obtained from a depth where the effective overburden pressure is about 100 kPa (about 5m). Indices obtained from other depths must be adjusted before using the charts. Fig. 10.2 indicates a correction factor, CN, based on the effective overburden stress at the depth where the actual SPT was performed. • The allowable bearing pressure determined from Figs. 10.1 and 10.2, are expected to produce settlements smaller than about 25 mm. SPT LIMITATIONS • The SPT is subject to many errors which affect the reliability of the SPT index, N. • Correlation between the SPT index and the internal friction angle of sand is very poor. Consequently, the calculation of allowable bearing pressure from N values should be considered with caution. • Nevertheless, in Canadian foundation engineering practice, footings are frequently designed using the SPT results, which indicates that the complete borehole information, not just the N value, can provide a reliable basis for sound engineering judgement. • The SPT index is not appropriate for determining the bearing pressure in fine-grained cohesive soils. EnvE 434 13 01/14/02 Bearing Capacity EnvE 434 14 01/14/02 Bearing Capacity What about foundation failure beneath a waste pile on a soft foundation? Let's examine how to estimate the maximum height of waste that may be placed on a soft foundation. H Waste pile Soft soil Failure Surface • Due to the inclined slope (solid line), calculation of the applied stress on the foundation soil is difficult and it does not lend itself to a simple bearing capacity solution. • In addition, we must consider the shear strength of the material in the waste pile that is being failed (as in a slope stability problem). • However, if the material is similar to municipal waste, although it has some shear strength, it requires considerable displacement to mobilize the strength. • By the time strength of the waste pile is mobilized, the foundation may have already failed. Thus for simplicity sake, at this point neglect any strength in the waste material. To initially overcome the geometry problem, we could conservatively estimate it as a vertical face, as shown with the dotted line. The allowable applied stress at the base of the soil would be equal to: qall = Using the CFEM formulation for qall, considering short-term loading (φ = 0), and solving for H, we obtain: H= The factor of safety here would be less than the 2.5 to 3.0 used in typical foundation design which is aimed at limiting displacements. An acceptable factor of safety would be 1.3 – 1.5. EnvE 434 15 01/14/02 Bearing Capacity EnvE 434 16 01/14/02 Bearing Capacity