bearing capacity2

D. P. Katale

Definitions of Bearing PressuresGross bearing pressure (qgross)Net bearing pressure (qnet)Effective gross bearing pressure (qgross)Effective net bearing pressure (qnet)Ultimate bearing capacity (qult)Safe bearing capacity (qs)Allowable bearing capacity (qa)Pressumed bearing capacity (qpressumed)Working Bearing pressure (qw )

Types of foundationShallow foundations sometimes called 'spread footings, include pads ('isolated footings'), strip footings and rafts. Deep foundations include piles, pile walls, diaphragm walls and caissonsCritereaD/B < 1shallowD/B > 6 deepD > 3 mdeep

Pad Foundation

Strip Foundation

AssumptionsRigid PlasticIsotropicHomogeniusUndrained, cu assumed constant throughout the failure zoneDrained c, f, g assumed constantSoil assumed weightlessImportant for drained conditions

Methods of Determining Bearing CapacityPressumed Bearing Capacity tablesBearing Capcity tables in building codesBased on past experienceTheoretical and semi-theoretical methodsField Test resultsSPTPLTCPT

Pressumed Bearing capacity (BS8004)

Failure modes

Failure modes of model footings

Factors affecting modes of failure Size of the foundationRelative density of the soil Permeability: relating to drained/undrained behaviourCompressibility: similar to RDShape: e.g. strips can only rotate one wayInteraction between adjacent foundations and other structuresRelative stiffnesses of soil and footing/structureIncidence and relative magnitude of horizontal loadings or momentsPresence of stiffer or weaker underlying layers.

Theortical and semi-theoretical methodsLimit Analysis methodsLower boundUpper boundLimit EquilibriumPrandtlSemi-circular slip mechanism Circular arc slip mechanism Skempton (1951)Terzaghi (1943)MeyerhofVesicHansen

Complete solutionBody, Surface loadsStresses

Strains

Displacements

Compatibility (geometry)EquilibriumConstitutives = EeLower boundUpper bound

Limit Analysis - Lower boundCollapse load calculated from;Statically admissible stress field,Satisfies boundary conditions,Is in equilibrium,No where violets the failure criterion,Is always a lower bound

Limit Analysis - Upper boundCollapse load calculated from;Kinematically admissible velocity field,For which the external rate of work done;Is greater than the internal dissipationIs always greater than the actual collapse loadAnd therefore an upper bound

Example Lower bound Bpo

Example Upper bound BqlowerBOYXOFXYFVelocity diagrame2Be2Bv2ve2ve2vvv

Semi-circular slip failure

Semi-circular slip failureMoment causing rotation = load x lever arm= [(qult x B] x [B] Moment resisting rotation = shear strength x length of arc x lever arm = [cu] x [p.B] x [B] + qoAt failure these are equal: qult x B x B = cu x p.B x B + qo Bearing pressure at failure , qult = 2p x shear strength + overburden pressure qult = 2pcu + qo (Ultimate bearing capacity)This is an upper-bound solution

Circular arc slip failure

Circular arc slip failureMoment causing rotation = load x lever arm = [qult x B ] x [B/2] Moment resisting rotation = shear strength x length of arc x lever arm = [cu] x [2a R] x [R] + qo At failure these are equal: qult x B x B/2 = cu x 2 a R x R + qo Since R = B / sin a : qult = cu x 4a /(sin a) + qo The worst case is when tana=2a at a = 1.1656 rad = 66.8 deg The ultimate bearing pressure at failure, qult is given byqult = 5.52 cu + qo

Plastic flow at failure-Ultimate bearing capacity

A strip footing

A strip footingA relatively undeformed wedge of soil below the foundation forms an active Rankine zone with angles (45 + f'/2). The wedge pushes soil outwards, causing passive Rankine zones to form with angles (45 - f'/2). The transition zones take the form of log spiral fans radius of the fan r = r0 .exp[q.tanf']. q is the fan angle in radians (between 0 and p/2) f' is the angle of friction of the soil ro = B/[2 cos(45+f'/2)] For purely cohesive soils (f = 0) the transition zones become circular

PrandltA relatively undeformed wedge of soil below the foundation forms an active Rankine zone with angles (45 + f'/2). The wedge pushes soil outwards, causing passive Rankine zones to form with angles (45 - f'/2). The transition zones take the form of log spiral fans For purely cohesive soils (f = 0) the transition zones become circular for which Prandtl had shown in 1920 that the solution is qfult = (2 + p) cu = 5.14 cu

Skemptonqult = cu .Ncu + qo where Ncu = Skempton's bearing capacity factor, given by the following equations: Ncu = Nc.sc.dc where sc is a shape factor and dc is a depth factor. Nc = 5.14 (Nq = 1, Ng = 0)sc = 1 + 0.2 (B/L) for B [ L dc = 1+ (0.053 Df /B ) for Df/B [ 4 Maximum values of dc when Df/B > 4 isdc = 1.459144(for strip, square and circular foundations)These equations can be presented as a chart.

Skemptons chartThese equations can be presented on a chart of Skempton's bearing capacity factor B/L = 0B/L = 1

Chart1

5.146.168

5.917.09

6.47.68

6.88.16

78.4

7.28.64

7.358.82

7.458.94

7.59

7.59

7.59

Strip

Square/circular

Depth to breadth ratio (D/B)

Bearing capacity factor (Ncu)

Skempton's Bearing capacity facotors

Sheet1

Skemptons Bearing Capacity factor Nc for clays

Bearing Capacity factor Nc

D/BStripSquare/circularrectangle

05.146.1685.1400

0.55.917.095.9083

16.47.686.4000

1.56.88.166.8000

278.47.0000

2.57.28.647.2000

37.358.827.3500

3.57.458.947.4500

47.597.5000

4.57.597.5000

57.597.5000

square or circular best fit 4th power series

-0.00950.128-0.71352.12496.168

Nc,rectangle = (0.84 + 0.16B/L)Nc,square

Nc,rectangle = (0.8333333 + 0.1666667B/L)Nc,square(for the limits 7.5 and 9.0 to convert correctly)

Determine Nc

DepthD =1

BreadthB =1

LengthL =10000000

Depth/breadth ratioD/B =1

Length/breadth ratioB/L =0.0000001

Nc6.431

Sheet1

00

00

00

00

00

00

00

00

00

00

00

Strip

Square/circular

Depth to breadth ratio (D/B)

Bearing capacity factor (Nc)

Skempton's Bearing capacity facotors

Sheet2

square orcircular

strip

Sheet3

Terzaghi (1943)qult = cNcsc + qoNqsq + gBNgsg Nc, Nq and Ng. = bearing capacity factors

B = width, L = length

sc sq sg square1.31.00.8circle1.31..00.6rectangle1+ 0.3(B/L) 1+ 0.2(B/L) 1 - 0.2(B/L)

Meryahofqult cNcscdc + qNqsqdq +0.5gBNgsgdgNq = etan f tan2(45+f/2)Nc = (Nq-1) cot fNg = (Nq-1) tan (1.4f)

sc = 1 + 0.2 Kp B/Lany fsq = sg = 1 + 0.1 Kp B/Lf >10osq = sg = 1f 10odq = dg = 1f

Hansen

The ultimate load which a foundation can support may be calculated using bearing capacity theory. For preliminary design, presumed bearing values can be used to indicate the pressures which would normally result in an adequate factor of safety. Alternatively, there is a range of empirical methods based on in situ test results. The ultimate bearing capacity (qult) is the value of bearing stress which causes a sudden catastrophic settlement of the foundation (due to shear failure). The allowable bearing capacity (qa) is the maximum bearing stress that can be applied to the foundation such that it is safe against instability due to shear failure and the maximum tolerable settlement is not exceeded. The allowable bearing capacity is normally calculated from the ultimate bearing capacity using a factor of safety (Fs). When excavating for a foundation, the stress at founding level is relieved by the removal of the weight of soil. The net bearing pressure (qnet) is the increase in stress on the soil. qn = q - qo qo = g D where D is the founding depth and g is the unit weight of the soil removed. The ultimate load which a foundation can support may be calculated using bearing capacity theory. For preliminary design, presumed bearing values can be used to indicate the pressures which would normally result in an adequate factor of safety. Alternatively, there is a range of empirical methods based on in situ test results. The ultimate bearing capacity (qult) is the value of bearing stress which causes a sudden catastrophic settlement of the foundation (due to shear failure). The allowable bearing capacity (qa) is the maximum bearing stress that can be applied to the foundation such that it is safe against instability due to shear failure and the maximum tolerable settlement is not exceeded. The allowable bearing capacity is normally calculated from the ultimate bearing capacity using a factor of safety (Fs). When excavating for a foundation, the stress at founding level is relieved by the removal of the weight of soil. The net bearing pressure (qnet) is the increase in stress on the soil. qn = q - qo qo = g D where D is the founding depth and g is the unit weight of the soil removed. Lower boundIf the soil is undergoing movement then it has failed. The forces that would stop this failure are due to the internal resistance and hence the internal dissipation of enerty by this internal resistance.If we determine the external energy rate that equates to the internal rate of energy dissipation then these external loads must be higher than the true collapse load and are therefore an upper bound.dc = 1+ sqrt(0.0527033Df /B ) for Df/B

bearing capacity2

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